Question 1

Angle of depression from 50 m lighthouse — horizontal distance From the top of a lighthouse 50 m above sea level, the angle of depression of an approaching boat is 30°. How far is the boat from the lighthouse foot along the sea surface?

Accepted Answer

Introduction / Context: Angle of depression equals the angle of elevation from the boat. With known vertical height, the horizontal distance follows from the tangent relation. Given Data / Assumptions: Height h = 50 m. Angle of depression θ = 30° ⇒ tan θ = h/d. Calm sea assumed; straight line of sight. Concept / Approach: Let d be the horizontal distance. Then tan 30° = h/d ⇒ d = h / tan 30°. Step-by-Step Solution: tan 30° = 1/√3d = 50 / (1/√3) = 50√3 m Verification / Alternative check: Numerically, √3 ≈ 1.732 ⇒ d ≈ 86.6 m, plausible for a 50 m elevation at 30°. Why Other Options Are Wrong: 25√3 and 25/√3 correspond to halving the height; 50/√3 would be tan 60°, not tan 30°. Common Pitfalls: Inverting the tan ratio or confusing angle of depression with angle of elevation magnitude. Final Answer: 50 √3 m

Question 2

Approaching a tower — elevation changes from 30° to 60° in 20 m On level ground, the angle of elevation of a tower is 30°. After walking 20 m toward the tower, the angle becomes 60°. Find the tower’s height.

Accepted Answer

Introduction / Context: Two observations from points along the same line to a tower give two tan equations in the same unknowns (height and the initial horizontal distance). Subtracting distances resolves both quickly. Given Data / Assumptions: Initial angle θ₁ = 30°, then θ₂ = 60° after moving 20 m closer. Let initial distance be x, tower height h. Level ground; vertical tower. Concept / Approach: tan 30° = h/x and tan 60° = h/(x − 20). Solve for x, then h. Use tan 30° = 1/√3 and tan 60° = √3. Step-by-Step Solution: h = x/√3 (from 30°)h = √3 (x − 20) (from 60°)Equate: x/√3 = √3(x − 20) ⇒ x = 3x − 60 ⇒ 2x = 60 ⇒ x = 30h = x/√3 = 30/√3 = 10√3 m Verification / Alternative check: At 30 m, tan 30° = h/x = (10√3)/30 = 1/√3; at 10 m (after moving 20), tan 60° = (10√3)/10 = √3. Checks out. Why Other Options Are Wrong: 20√3 is double the correct height; 10(√3 − 1) and “None” do not satisfy both tan equations. Common Pitfalls: Using 20 as the new distance (x − 20 = 20) without solving, or mixing sine/cosine for tangent relations. Final Answer: 10 √3 m

Question 3

Height & Distance — two boats due east with different angles of depression: From the top of a lighthouse, the angles of depression to two boats lying due east are 45° (nearer boat) and 30° (farther boat). The boats are 60 m apart along the same stra…

Accepted Answer

Introduction / Context: Problems on angles of depression/elevation reduce to right triangles where tan(angle) = height / horizontal distance. With two sightlines from the same point, we can relate two horizontal distances to the same unknown height and then use their given separation to solve for height. Given Data / Assumptions: Angle to nearer boat = 45°; angle to farther boat = 30°. Horizontal separation between boats = 60 m, collinear and due east of the lighthouse base. Let the lighthouse height be h meters; all ground is level. Concept / Approach: Let d1 be horizontal distance to the 45° boat, and d2 to the 30° boat. Using tan θ = h / d, we have d1 = h / tan45° = h and d2 = h / tan30° = h√3. The boats lie on the same ray, so d2 − d1 = 60. Step-by-Step Solution: d2 − d1 = h√3 − h = 60h(√3 − 1) = 60h = 60 / (√3 − 1) = 60(√3 + 1) / (3 − 1) = 30(√3 + 1) Verification / Alternative check: Numerically, √3 ≈ 1.732 ⇒ h ≈ 30 × 2.732 ≈ 81.96 m. Then d1 ≈ 81.96 m and d2 ≈ 141.96 m, whose difference is ≈ 60 m, matching the condition. Why Other Options Are Wrong: 60√3 ≈ 103.92 m and 30(√3 − 1) ≈ 21.96 m do not satisfy d2 − d1 = 60 for consistent tangents. “90” is also inconsistent with the trigonometric relations. Common Pitfalls: Using d1 + d2 instead of the difference; confusing tan with cot; forgetting that angle of depression equals the angle of elevation at the ground point. Final Answer: 30 ( √3 + 1 )

Question 4

Flagstaff shadow — find the Sun’s angle with the ground: A vertical flagstaff of height 6 m casts a shadow of length 2√3 m on level ground. What is the angle of elevation of the Sun (in degrees)?

Accepted Answer

Introduction / Context: Shadow questions model a right triangle with height as the opposite side and shadow as the adjacent side; thus tan(θ) = height / shadow. Using exact radical values allows a clean angle result. Given Data / Assumptions: Height = 6 m Shadow = 2√3 m Ground is horizontal; flagstaff is vertical. Concept / Approach: Compute tan θ and recognize a standard angle whose tangent equals √3. Step-by-Step Solution: tan θ = 6 / (2√3) = 3 / √3 = √3Therefore, θ = 60° (since tan 60° = √3) Verification / Alternative check: Using numeric √3 ≈ 1.732, tan θ ≈ 1.732 ⇒ θ ≈ 60°; consistent with standard trigonometric values. Why Other Options Are Wrong: At 30°, tan = 1/√3; at 45°, tan = 1; at 75°, tan is much larger than √3. “None” is unnecessary since an exact match exists. Common Pitfalls: Inverting the ratio (shadow/height) and using cot instead of tan; mixing degrees and radians. Final Answer: 60°

Question 5

Similar-triangles under the same Sun — pole and man comparison: A vertical pole 6 m high casts an 8 m shadow. At the same time and place, a man nearby casts a 2.4 m shadow. What is the height of the man (in meters)?

Accepted Answer

Introduction / Context: When two vertical objects cast shadows at the same time, similar triangles apply: height/shadow is constant. We can scale the man’s shadow by the known ratio from the pole to find the man’s height. Given Data / Assumptions: Pole height = 6 m; its shadow = 8 m Man’s shadow = 2.4 m; man’s height = ? Same Sun (same solar elevation), level ground. Concept / Approach: Use height/shadow = constant for both objects (similar right triangles). Thus hman / 2.4 = 6 / 8. Step-by-Step Solution: hman = 2.4 × (6/8) = 2.4 × 0.75 = 1.8 m Verification / Alternative check: Both give the same tangent ratio: 6/8 = 1.8/2.4 = 0.75, confirming the similarity scaling is correct. Why Other Options Are Wrong: Values 1.4, 1.6, 2.0, or 1.5 m do not preserve the same height-to-shadow ratio with a 2.4 m shadow. Common Pitfalls: Inverting the ratio (8/6 instead of 6/8); mixing units (they are already in meters). Final Answer: 1.8 m

Question 6

Boat recedes from a tower — angle of depression changes from 45° to 30° in 5 s: A man at the top of a tower observes a boat. When the boat is 60 m from the base, the angle of depression is 45°. After 5 seconds, the angle of depression becomes 30°. A…

Accepted Answer

Introduction / Context: Changing angles of depression from the same height translate into changing horizontal distances. With height known from one sighting, the second sighting gives a farther horizontal distance. The difference over time gives horizontal speed. Given Data / Assumptions: At first: horizontal distance x1 = 60 m; angle = 45° ⇒ tower height h = x1 × tan45° = 60 m. After 5 s: angle = 30° ⇒ horizontal distance x2 = h / tan30° = 60√3 m. Straight-line motion away from the tower, still water. Concept / Approach: Speed v = (x2 − x1)/time. Convert m/s to km/h by multiplying by 3.6. Step-by-Step Solution: x1 = 60 m; x2 = 60√3 ≈ 103.92 mΔx ≈ 103.92 − 60 = 43.92 m in 5 s ⇒ v ≈ 43.92/5 = 8.784 m/sIn km/h: 8.784 × 3.6 ≈ 31.62 km/h ≈ 32 km/h (rounded) Verification / Alternative check: Reversing: 32 km/h ≈ 8.888… m/s; over 5 s ≈ 44.44 m; close to Δx ≈ 43.92 m — consistent with rounding. Why Other Options Are Wrong: 36, 38, 42 km/h overshoot the computed horizontal speed; 30 km/h undershoots; 32 km/h is the nearest correct approximation. Common Pitfalls: Using heights instead of horizontal distances; mixing degrees with radians; forgetting unit conversion to km/h. Final Answer: 32 Km/h

Question 7

Rectangle with midpoint and triangle at P — evaluate sin(∠CPB): In rectangle ABCD, AB : BC = 3 : 2. Point P is the midpoint of AB. What is the value of sin(∠CPB)?

Accepted Answer

Introduction / Context: Coordinate geometry streamlines trig in plane figures. Placing the rectangle on axes lets us compute vectors for the sides that define ∠CPB and then evaluate sin using the cross-product formula for the sine of the angle between two vectors. Given Data / Assumptions: AB : BC = 3 : 2; take convenient lengths AB = 3 and BC = 2 (units). Let A = (0, 0), B = (3, 0), C = (3, 2), D = (0, 2). P is the midpoint of AB ⇒ P = (1.5, 0). Concept / Approach: At P, form vectors to B and C: PB and PC. For vectors u and v, sin θ = |u × v| / (|u||v|), where in 2D the cross magnitude is |xuyv − yuxv|. Step-by-Step Solution: PB = B − P = (3 − 1.5, 0 − 0) = (1.5, 0)PC = C − P = (3 − 1.5, 2 − 0) = (1.5, 2)|PB| = √(1.5² + 0²) = 1.5; |PC| = √(1.5² + 2²) = √6.25 = 2.5|PB × PC| = |1.5·2 − 0·1.5| = 3sin(∠CPB) = 3 / (1.5 × 2.5) = 3 / 3.75 = 0.8 = 4/5 Verification / Alternative check: Using dot/cross formulas yields the same 0.8. Ratio choice does not affect result because any positive scalar multiple keeps the angle unchanged. Why Other Options Are Wrong: 3/5 (0.6), 2/5 (0.4), 3/4 (0.75), and 1/2 (0.5) do not match the computed 0.8. Common Pitfalls: Using length ratio AB/BC directly inside trigonometric functions or mixing up the vectors’ order (which flips the sign but not the magnitude for sine). Final Answer: 4 5

Question 8

Complementary angles from two points a and b — find the tower height: From two points P and Q on the same straight line through the foot of a tower, at distances a and b from the base, the angles of elevation of the top are complementary. What is th…

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Introduction / Context: Complementary angles (θ and 90° − θ) imply tan(θ) · tan(90° − θ) = 1. With the same tower height h and two horizontal distances a and b, we can connect the tangents and obtain a simple closed-form for h. Given Data / Assumptions: Point P at distance a; angle of elevation = θ. Point Q at distance b; angle of elevation = 90° − θ. Same height h and level ground. Concept / Approach: Write tan θ = h/a and tan(90° − θ) = cot θ = a/h. But from Q we also have tan(90° − θ) = h/b ⇒ equate and use tan θ · tan(90° − θ) = 1 to relate a, b, and h. Step-by-Step Solution: From P: tan θ = h/aFrom Q: tan(90° − θ) = h/b = cot θ = a/hSo h/b = a/h ⇒ h² = ab ⇒ h = √ab (height is positive) Verification / Alternative check: Then tan θ = h/a = √ab / a = √(b/a) and tan(90° − θ) = h/b = √ab / b = √(a/b). Their product is 1, confirming complementarity. Why Other Options Are Wrong: a·b, a b (ambiguous), a²b², or a + b do not satisfy the trig identities for complementary angles and the geometry here. Common Pitfalls: Forgetting tan(90° − θ) = cot θ; attempting to add distances instead of using product relations. Final Answer: √ab

Question 9

Shadow equals (1/√3) of the height — find the Sun’s elevation angle: The length of the shadow of a vertical tower is 1/√3 times its height. What is the angle of elevation of the Sun?

Accepted Answer

Introduction / Context: Angle of elevation θ relates height h and shadow s by tan θ = h/s. If s is a fixed fraction of h, tan θ is a fixed constant leading to a standard angle result. Given Data / Assumptions: Shadow length s = (1/√3) × h. Vertical tower; level ground. Concept / Approach: Compute tan θ = h/s = h / (h/√3) = √3, then identify θ from the exact trig value. Step-by-Step Solution: tan θ = √3 ⇒ θ = 60° Verification / Alternative check: At 60°, tan = √3; plugging back yields s = h/√3 as stated. The geometry is consistent. Why Other Options Are Wrong: 30° gives tan = 1/√3; 45° gives tan = 1; 90° is vertical and produces no finite shadow; 15° is much smaller. Common Pitfalls: Using cot instead of tan or misreading “1 √3 times” as √3 times. Here it is explicitly 1/√3 of the height. Final Answer: 60 °

Question 10

An object moves with a constant speed of 5 feet per second. How many feet will it travel in 1 hour of continuous motion?

Accepted Answer

Introduction: This is a basic speed–time–distance question involving unit conversion from seconds to hours. It tests your ability to correctly convert time units and apply the fundamental relationship between distance, speed and time. Given Data / Assumptions: Speed of the object = 5 feet per second. Time of travel = 1 hour. The motion is assumed to be uniform (constant speed). We must calculate the total distance travelled in feet. Concept / Approach: The core formula relating distance, speed and time is: Distance = Speed * TimeHowever, the units must be consistent. Here, speed is in feet per second, while time is given in hours. We first convert hours to seconds, then multiply by the speed. Step-by-Step Solution: Step 1: Convert time from hours to seconds.1 hour = 60 minutes.Each minute = 60 seconds.Therefore, total time in seconds = 60 * 60 = 3600 seconds.Step 2: Use the distance formula.Distance = Speed * TimeDistance = 5 feet per second * 3600 secondsStep 3: Multiply.Distance = 5 * 3600 = 18000 feet Verification / Alternative check: We can break the hour into smaller chunks to verify. In 1 second, the object travels 5 feet. In 10 seconds, it travels 50 feet. In 100 seconds, 500 feet. In 1000 seconds, 5000 feet. For 3600 seconds (just over 3.5 times 1000), 5 * 3600 again gives 18000 feet, confirming the calculation. Why Other Options Are Wrong: 3000 and 1800: These correspond to using 10 minutes or 6 minutes instead of 1 hour, indicating incorrect time conversion. 15000 and 12000: These are improperly scaled values, likely from miscalculating 5 * 3000 or using 50 minutes instead of 60 minutes. Common Pitfalls: The most common mistake is forgetting that 1 hour equals 3600 seconds, not 3000 or any rounded figure. Another error is mixing up units (e.g., treating 5 feet per second as 5 feet per minute). Always standardize units before applying formulas. Final Answer: In 1 hour, the object travels a total distance of 18000 feet.

Question 11

A vertical toy that is 18 cm high casts a shadow 8 cm long on the ground. At the same time and under the same lighting conditions, a vertical pole casts a shadow 48 m long on the ground. What is the height of the pole?

Accepted Answer

Introduction: This question involves the concept of similar triangles formed by objects and their shadows when illuminated by the same light source, such as the sun. It is a standard height-and-distance problem used in aptitude exams to test proportional reasoning. Given Data / Assumptions: Height of the toy = 18 cm. Length of the toy's shadow = 8 cm. Length of the pole's shadow = 48 m. Both toy and pole are vertical and illuminated under the same conditions, so the angles of elevation of the light source are equal. We must find the height of the pole. Concept / Approach: When two vertical objects cast shadows under the same lighting, the triangles formed by each object and its shadow are similar. For similar triangles, the ratio of corresponding sides (height to shadow) is equal. Height of toy / Shadow of toy = Height of pole / Shadow of poleWe must also handle unit consistency, converting all measurements to the same unit if needed. Step-by-Step Solution: Step 1: Write the proportional relationship.18 cm / 8 cm = Height of pole / 48 mStep 2: Use the ratio 18 / 8.18 / 8 = 9 / 4Step 3: Express height of pole in metres.Height of pole = (18 / 8) * 48 mStep 4: Simplify stepwise.(18 / 8) * 48 = (9 / 4) * 48(9 / 4) * 48 = 9 * 12 = 108Therefore, height of pole = 108 m Verification / Alternative check: The height to shadow ratio for the toy is 18 : 8 which simplifies to 9 : 4. For the pole, with height 108 m and shadow 48 m, the ratio is 108 : 48, which simplifies by dividing both numbers by 12: 108 ÷ 12 = 9 and 48 ÷ 12 = 4. So the ratio is also 9 : 4, confirming similarity and correctness. Why Other Options Are Wrong: 1080 cm (10.8 m): This does not preserve the correct ratio with the shadow length 48 m. 180 m and 96 m: These values give a different height-to-shadow ratio than 9 : 4, so they are inconsistent with similar triangles. 118 cm: This is far too small relative to a 48 m shadow and does not fit the proportion at all. Common Pitfalls: Students often mix up units (centimetres and metres) or forget to convert properly. Another mistake is using an incorrect ratio such as shadow/height instead of height/shadow. Staying consistent with ratios and units avoids these issues. Final Answer: The height of the pole is 108 m.

Question 12

A man standing at point P views the top of a tower at an angle of elevation of 30°. He then walks some distance towards the tower, after which the angle of elevation of the top becomes 60°. What is the distance between the base of the tower and poin…

Accepted Answer

Introduction: This is a height-and-distance problem involving angles of elevation. However, unlike typical questions that provide enough information to compute an exact distance, this one is designed to test whether you can identify when the given data is insufficient to determine a unique answer. Given Data / Assumptions: The man initially stands at point P and sees the top of a tower at an angle of elevation of 30°. He walks some (unspecified) distance towards the tower and then sees the top at an angle of elevation of 60°. No information is given about the distance he walks or the height of the tower. We are asked for the distance between the base of the tower and point P. Concept / Approach: We typically use right-angled trigonometry for such problems. Let the height of the tower be h and the original horizontal distance from P to the base be d. After walking a distance s towards the tower, the new distance becomes d − s. The angles give us two tangent relations: tan(30°) = h / dtan(60°) = h / (d − s)However, we have three unknowns (h, d and s) and only two equations. Step-by-Step Solution: Step 1: From the first position.tan(30°) = h / d ⇒ h = d * tan(30°)Step 2: From the second position.tan(60°) = h / (d − s)Step 3: Substitute h from Step 1.tan(60°) = (d * tan(30°)) / (d − s)Step 4: Observe unknowns.We now have one equation involving d and s, but there are two unknowns and no additional condition, so infinitely many (d, s) pairs satisfy the equation. Verification / Alternative check: If we choose any specific height h and initial distance d that satisfy tan(30°) = h / d, we can always find an s such that tan(60°) = h / (d − s). Many combinations are possible, meaning the original distance d is not uniquely determined by the given information. Why Other Options Are Wrong: 8 units, 12 units, 10 units: Each of these gives a specific numerical distance, but there is no data to justify any of these unique values. They are arbitrary guesses. None of these: This usually means there is some other definite numeric answer, which is not the case here. The real issue is that the data itself is insufficient. Common Pitfalls: Many students try to force a solution by assuming a convenient value for the height or distance without justification. Recognizing data insufficiency is an important skill and often intentionally tested in such questions. Final Answer: The problem does not provide enough information to determine the original distance uniquely, so the correct answer is Data inadequate.

Question 13

Jack jogs around a race course that is 3 miles long. He takes 20 minutes to complete the first lap and 25 minutes to complete the second lap. What is his average speed in miles per hour for the entire jog?

Accepted Answer

Introduction: This question is about average speed over a trip where the speeds (or times) over different segments are not the same. It is important to remember that average speed is total distance divided by total time, not the average of individual speeds or times. Given Data / Assumptions: Length of one lap around the course = 3 miles. Time for first lap = 20 minutes. Time for second lap = 25 minutes. Jack runs two laps in total. We must find his average speed in miles per hour. Concept / Approach: Average speed is always: Average speed = Total distance / Total timeWe must be careful to convert total time into hours because the answer is required in miles per hour (mph). Step-by-Step Solution: Step 1: Compute total distance.Each lap is 3 miles, and he runs 2 laps.Total distance = 2 * 3 = 6 miles.Step 2: Compute total time in minutes.Total time = 20 minutes + 25 minutes = 45 minutes.Step 3: Convert total time to hours.45 minutes = 45 / 60 hours = 0.75 hours.Step 4: Apply the average speed formula.Average speed = Total distance / Total timeAverage speed = 6 miles / 0.75 hours = 8 mph Verification / Alternative check: You can check by proportion: if he travelled 8 miles in 1 hour, then in 0.75 hours he would travel 8 * 0.75 = 6 miles, which matches the actual total distance. This confirms that 8 mph is consistent with the given data. Why Other Options Are Wrong: 6 mph: Would imply a distance of 6 * 0.75 = 4.5 miles in 0.75 hours, which is too small. 9 mph: In 0.75 hours, 9 mph would cover 6.75 miles, larger than 6 miles. 10 mph: Would correspond to 7.5 miles in 0.75 hours, again incorrect. 7 mph: In 0.75 hours, distance would be 5.25 miles, not 6 miles. Common Pitfalls: Students sometimes average times or incorrectly average hypothetical speeds per lap. The correct method is always to sum total distance and total time first, then divide. This is critical when segment times or speeds differ. Final Answer: Jack's average speed for the whole jog is 8 mph.

Question 14

From the top of a hill that is 100 m high, the top and bottom of a nearby vertical tower are seen at angles of depression of 30° and 60° respectively. What is the height of the tower?

Accepted Answer

Introduction: This is a trigonometry-based height-and-distance question that uses angles of depression from the top of a hill to the top and bottom of a tower. It tests your understanding of right-angled triangles, angles of depression and the use of tangent functions in solving vertical height problems. Given Data / Assumptions: Height of the hill = 100 m. Angle of depression to the top of the tower = 30°. Angle of depression to the bottom (base) of the tower = 60°. The hill and the tower stand on the same horizontal ground. We must find the height of the tower. Concept / Approach: Angles of depression from the horizontal at the observer equal the corresponding angles of elevation from the object to the observer. We let the horizontal distance from the hill top to the base of the tower be d, and the height of the tower be H. Using tangent in right-angled triangles: tan(60°) = vertical drop from hill top to tower base / dtan(30°) = vertical drop from hill top to tower top / d Step-by-Step Solution: Step 1: Vertical distances.Hill top is 100 m above ground; tower base is at ground level.Thus, vertical drop to base = 100 m.Vertical drop to top of tower = 100 − H (since tower top is H m above ground).Step 2: Use tan(60°) for base.tan(60°) = 100 / d ⇒ √3 = 100 / d ⇒ d = 100 / √3Step 3: Use tan(30°) for top.tan(30°) = (100 − H) / dBut tan(30°) = 1 / √3 and d = 100 / √3.So, 1 / √3 = (100 − H) / (100 / √3)Step 4: Simplify the equation.1 / √3 = (100 − H) * (√3 / 100)Multiply both sides by √3: 1 = 3 * (100 − H) / 100100 = 3 * (100 − H)100 = 300 − 3H ⇒ 3H = 200 ⇒ H = 200 / 3 ≈ 66.6 m Verification / Alternative check: With H ≈ 66.6 m, vertical drop to top is 100 − 66.6 ≈ 33.4 m. Then tan(30°) ≈ 0.577 for 33.4 / d, and tan(60°) ≈ 1.732 for 100 / d. With d = 100 / 1.732 ≈ 57.7 m, both ratios align closely with the standard tangent values, confirming the result. Why Other Options Are Wrong: 42.2 m, 33.45 m, 58.78 m, 50 m: Substituting any of these heights into the tangent relationships will not satisfy both angle conditions simultaneously. Only H ≈ 66.6 m maintains the correct geometry for both 30° and 60° angles of depression. Common Pitfalls: Students may mix up which angle corresponds to the top or bottom, or forget that angles of depression equal angles of elevation. Another frequent error is taking vertical distances incorrectly, for example using H instead of 100 − H for the top of the tower. Final Answer: The height of the tower is approximately 66.6 m.

Question 15

A flagstaff 17.5 m high casts a shadow 40.25 m long. Under the same conditions, a building casts a shadow 28.75 m long. What is the height of the building?

Accepted Answer

Introduction: This question is a classic application of similar triangles in height-and-distance problems. The heights of vertical objects and lengths of their shadows are proportional when the angle of elevation of the light source remains the same, such as under constant sunlight. Given Data / Assumptions: Height of the flagstaff = 17.5 m. Length of the flagstaff's shadow = 40.25 m. Length of the building's shadow = 28.75 m. The flagstaff and the building are vertical and exposed to the same light source at the same time. We must determine the height of the building. Concept / Approach: When two vertical objects cast shadows under identical lighting, triangles formed by each object and its shadow are similar. This implies: Height of flagstaff / Shadow of flagstaff = Height of building / Shadow of buildingWe use this proportion to compute the unknown height. Step-by-Step Solution: Step 1: Set up the proportion.17.5 / 40.25 = Height of building / 28.75Step 2: Express height of building H.H = 28.75 * (17.5 / 40.25)Step 3: Compute numerically.First, notice 17.5 / 40.25 is a constant ratio.Using direct calculation, H = (28.75 * 17.5) / 40.25This evaluates to H = 12.5 m Verification / Alternative check: Compare the ratios. For the flagstaff, height : shadow = 17.5 : 40.25. For the building, using H = 12.5 m and shadow 28.75 m, ratio = 12.5 : 28.75. Both ratios simplify to the same value numerically, confirming similarity of triangles and the correctness of H = 12.5 m. Why Other Options Are Wrong: 14 m, 13.5 m, 11.4 m, 16 m: Substituting any of these values into the proportion H / 28.75 = 17.5 / 40.25 yields a mismatch; the ratios of height to shadow would not be equal to that of the flagstaff. Only 12.5 m preserves the proportional relationship required by similar triangles. Common Pitfalls: Typical mistakes include using the inverse ratio (shadow/height) incorrectly or not maintaining consistent units. Another error is rough mental approximation without checking that both ratios are equal. Writing the proportion carefully and performing stepwise calculation prevents such errors. Final Answer: The height of the building is 12.5 m.

Question 16

In a right-angled triangle model of a ladder leaning against a vertical wall, the angle of elevation between the ladder and the horizontal ground is 60°. If the foot of the ladder is placed 4.6 m away from the wall along the ground, what is the full…

Accepted Answer

Introduction: This question is a classic application of trigonometry in height and distance problems. The situation, a ladder leaning against a wall, forms a right-angled triangle where the ladder is the hypotenuse. We are given the horizontal distance from the wall and the angle the ladder makes with the ground, and we must find the ladder's length. Given Data / Assumptions: The ladder leans against a vertical wall. The angle between the ladder and the ground is 60°. The horizontal distance from the foot of the ladder to the wall is 4.6 m. The wall is perfectly vertical and the ground is horizontal, forming a right angle at the base. Concept / Approach: In a right-angled triangle, the cosine of an acute angle is defined as adjacent side divided by hypotenuse. Here: cos(60°) = (horizontal distance) / (ladder length).We know cos(60°) = 1/2. So we can rearrange the formula to find the ladder length as: ladder length = (horizontal distance) / cos(60°). Step-by-Step Solution: Step 1: Identify sides: adjacent side = 4.6 m, hypotenuse = ladder length (L).Step 2: Use the cosine relation: cos(60°) = 4.6 / L.Step 3: Substitute cos(60°) = 1/2, so 1/2 = 4.6 / L.Step 4: Rearrange to find L: L = 4.6 / (1/2) = 4.6 * 2.Step 5: Calculate: L = 9.2 m. Verification / Alternative check: If the ladder is 9.2 m long, then the horizontal projection should be 9.2 * cos(60°) = 9.2 * 0.5 = 4.6 m, which matches the given distance. This confirms that the calculation is consistent. Why Other Options Are Wrong: 4.6 m would mean the ladder lies flat on the ground, not reaching the wall above ground level. Values like 2.3 m, 6.4 m, or 7.8 m are all shorter than the hypotenuse required by a 60° triangle with a 4.6 m base. None of them satisfy the cosine relation for 60° with the given adjacent side. Common Pitfalls: Students sometimes mistakenly use sine instead of cosine, or treat the 4.6 m as the hypotenuse. Remember: the ladder is the slant side (hypotenuse), the ground distance is adjacent to the angle with the ground, so cosine is the correct trigonometric ratio to use. Final Answer: The length of the ladder is 9.2 m.

Question 17

A ladder is resting against a vertical wall, and initially the bottom of the ladder is 2.5 m away from the wall along the ground. If the ladder slips so that its top slides 0.8 m down the wall and, at the same time, the bottom moves 1.4 m farther aw…

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Introduction: This problem uses right-angled triangles and the Pythagoras theorem to determine the fixed length of a ladder as its position changes. The ladder slides while maintaining the same length, giving us two different right triangles with the same hypotenuse. By equating the two expressions for the ladder length, we can solve for its value. Given Data / Assumptions: Initially, distance from wall to foot of ladder along the ground = 2.5 m. The top of the ladder is at some height h above the ground. The ladder then slips 0.8 m down, so new height = h − 0.8 m. The bottom moves 1.4 m away, so new horizontal distance = 2.5 + 1.4 = 3.9 m. The ladder length remains constant throughout. Concept / Approach: We model both situations as right-angled triangles. If L is the ladder length, then: L² = 2.5² + h² (initial position)L² = 3.9² + (h − 0.8)² (after slipping)Since L² is the same, we equate these two expressions and solve for h, then back-calculate L. Step-by-Step Solution: Step 1: Write the first equation: L² = 2.5² + h² = 6.25 + h².Step 2: Write the second equation: L² = 3.9² + (h − 0.8)² = 15.21 + h² − 1.6h + 0.64.Step 3: Simplify the second: L² = 15.85 + h² − 1.6h.Step 4: Set the two expressions equal: 6.25 + h² = 15.85 + h² − 1.6h.Step 5: Cancel h² from both sides, giving 6.25 = 15.85 − 1.6h.Step 6: Rearrange: −9.6 = −1.6h, so h = 9.6 / 1.6 = 6 m.Step 7: Use the initial triangle to find L: L² = 6.25 + 6² = 6.25 + 36 = 42.25.Step 8: Therefore, L = √42.25 = 6.5 m. Verification / Alternative check: Check with the second position: L² should also equal 3.9² + (6 − 0.8)² = 15.21 + 5.2² = 15.21 + 27.04 = 42.25, giving L = 6.5 m again. This confirms consistency. Why Other Options Are Wrong: Values like 6.2 m, 6.8 m, 7.5 m, or 5.9 m do not satisfy both right-triangle configurations simultaneously. When substituted, they fail to give the same L for both positions, violating the fixed ladder-length condition. Common Pitfalls: Common errors include forgetting that the ladder length is constant, mishandling the algebra when expanding (h − 0.8)², or rounding too early. Carefully expanding the square and equating both expressions for L² ensures an accurate answer. Final Answer: The length of the ladder is 6.5 m.

Question 18

A right circular cone has a vertical height of 24 cm and a slant height of 25 cm. Using π = 22/7, what is the curved surface area (in cm²) of this cone?

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Introduction: This question applies basic solid geometry, specifically the formula for the curved surface area of a right circular cone. You are given the height and slant height, and are asked to compute the lateral (curved) area using a standard formula and a specified value of π. Given Data / Assumptions: Height of the cone, h = 24 cm. Slant height of the cone, l = 25 cm. π is to be taken as 22/7. The cone is right circular, so its axis is perpendicular to the base. Concept / Approach: The curved surface area (CSA) of a right circular cone is given by: CSA = π * r * lwhere r is the radius of the base, and l is the slant height. Here, we are given h and l. We first find r using the Pythagoras theorem in the right triangle formed by radius, height, and slant height. Step-by-Step Solution: Step 1: Use the relation l² = r² + h².Step 2: Substitute values: 25² = r² + 24².Step 3: Compute squares: 625 = r² + 576.Step 4: Rearrange: r² = 625 − 576 = 49.Step 5: Take the square root: r = 7 cm.Step 6: Now apply the CSA formula: CSA = π * r * l = (22/7) * 7 * 25.Step 7: Simplify: (22/7) * 7 = 22, so CSA = 22 * 25 = 550 cm². Verification / Alternative check: You can quickly verify by checking the Pythagoras relation again: 7² + 24² = 49 + 576 = 625 = 25². This confirms that r = 7 cm is correct and that the cone dimensions are consistent. Why Other Options Are Wrong: Values such as 572 cm², 528 cm², 539 cm², or 600 cm² arise from incorrect radius calculations or from misusing the CSA formula. Only 550 cm² matches the correct evaluation of π * r * l with π = 22/7, r = 7 cm, and l = 25 cm. Common Pitfalls: Students sometimes use π * r² (which is the area of a circle) instead of π * r * l for the curved surface area of a cone, or mistakenly use height instead of slant height in the formula. Ensuring you use the correct geometrical formula is crucial for such problems. Final Answer: The curved surface area of the cone is 550 cm².

Question 19

Two men stand on the same side of a vertical pillar that is 75 m high and observe the angle of elevation of the top of the pillar to be 30° and 60°, respectively. What is the distance (in metres) between the two men?

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Introduction: This is a standard height and distance problem using trigonometry. Two observers stand on the same side of a vertical pillar and see its top under different angles of elevation. Using tangent relations for right-angled triangles, we can find each man’s horizontal distance from the pillar and then determine the distance between them. Given Data / Assumptions: Height of the pillar = 75 m. First man's angle of elevation = 60°. Second man's angle of elevation = 30°. Both are on the same straight line on one side of the pillar. The ground is level and the pillar is perfectly vertical. Concept / Approach: For a vertical object of height H and a point on level ground at distance d from its foot, the angle of elevation θ satisfies: tan(θ) = H / d.Using this, we can find the distances of each man from the pillar. The man with the larger angle (60°) is closer; the man with the smaller angle (30°) is farther. The difference of these distances gives the distance between them. Step-by-Step Solution: Step 1: Let d₁ be the distance of the man seeing 60°, and d₂ be the distance of the man seeing 30°.Step 2: Using tan(60°) = √3: √3 = 75 / d₁ ⇒ d₁ = 75 / √3 = 25√3 m.Step 3: Using tan(30°) = 1/√3: 1/√3 = 75 / d₂ ⇒ d₂ = 75√3 m.Step 4: Distance between the two men = d₂ − d₁ = 75√3 − 25√3 = 50√3 m. Verification / Alternative check: If 50√3 ≈ 50 * 1.732 = 86.6 m, then one man is at about 43.3 m from the pillar and the other at about 129.9 m from the pillar, a reasonable configuration given the two angles 60° (steeper) and 30° (shallower). This matches our trigonometric expectations. Why Other Options Are Wrong: Distances like 25√3 m or 75√3 m represent individual distances from the pillar, not the separation between the men. Purely numeric values such as 75 m or 100 m do not satisfy the tangent-based relations for both angles simultaneously. Common Pitfalls: Some learners incorrectly add instead of subtracting distances or swap the distances corresponding to 30° and 60°. Remember: a larger angle of elevation means a shorter horizontal distance to the object. Final Answer: The distance between the two men is 50√3 m.

Question 20

Two parallel streets run north–south. A person walks north along the first street and wants to reach the second street, which lies somewhere to his right (east). At one point, he turns 150° to the right and walks for 15 minutes at 20 km/h. Then he t…

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Introduction: This is a challenging direction and distance problem involving vectors and angles. A person changes direction twice and eventually reaches a parallel street. The key is to resolve each leg of the journey into east–west (horizontal) and north–south (vertical) components and then use these to find the separation between the two parallel streets. Given Data / Assumptions: The two streets are straight, parallel, and oriented north–south. Initial direction: the person walks north along the first street. First turn: 150° to the right from north, then walks 15 minutes at 20 km/h. Second turn: 60° to the left from his new direction, then walks 20 minutes at 30 km/h. He ends up exactly on the second street. Concept / Approach: We choose a coordinate system: north as positive y-axis and east as positive x-axis. The streets are vertical (north–south) lines, so the separation between them is the net east–west (x) displacement from the starting point to the final position. We compute the displacement vectors for each leg using speed × time for distance and trigonometry for direction. Step-by-Step Solution: Step 1: Distance of first leg: 20 km/h * (15/60) h = 5 km.Step 2: From north, a right turn of 150° gives a bearing 150° clockwise from north, equivalent to a direction 60° below east (i.e., 60° south of east).Step 3: Components of first leg: x₁ = 5 * cos 60° = 5 * 0.5 = 2.5 km east; y₁ = 5 * (−sin 60°) = 5 * (−√3/2) ≈ −4.33 km (southward).Step 4: Distance of second leg: 30 km/h * (20/60) h = 10 km.Step 5: A left turn of 60° from this heading brings him to due east (90° bearing), so the second leg is purely eastward.Step 6: Components of second leg: x₂ = 10 km east; y₂ = 0 km.Step 7: Total eastward displacement: x = x₁ + x₂ = 2.5 + 10 = 12.5 km.Step 8: Total north–south displacement is irrelevant for street separation; only east–west displacement matters. Verification / Alternative check: Because both streets are parallel and oriented north–south, any point on each street has the same x-coordinate along that street. The person starts on the first street (x = 0) and ends on the second street (x = 12.5 km), so the horizontal separation must be 12.5 km. This confirms that 12.5 km is the correct distance between the streets. Why Other Options Are Wrong: Distances like 7.5 km, 10.5 km, or 15 km arise if you incorrectly project only one leg or take total path length. The problem asks for perpendicular separation between streets, not total distance walked. 9.5 km does not correspond to any correct component calculation based on the given speeds, times, and angles. Common Pitfalls: Typical mistakes involve misinterpreting the 150° right turn (confusing it with 30°), or forgetting that only the east–west displacement gives the distance between parallel north–south streets. Careful diagramming and consistent coordinate axes are essential for such vector-based problems. Final Answer: The distance between the two streets is 12.5 km.

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