The perimeter of the triangular base of a right prism is 60 cm. The sides of the base are in the ratio 5 : 12 : 13. If the prism height is 60 cm, find the volume of the prism (in cm^3).
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A7200 cm3
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B6000 cm3
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C6600 cm3
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D5400 cm3
Answer
Correct Answer: 7200 cm3
Explanation
Introduction / Context:For a right prism, the volume equals the area of the base times the prism height. The 5:12:13 ratio indicates a Pythagorean triple (right triangle), simplifying the base area calculation.
Given Data / Assumptions:
- Perimeter = 60 cm; ratio 5:12:13 → sides 10 cm, 24 cm, 26 cm.
- Right triangle legs: 10 cm and 24 cm; hypotenuse 26 cm.
- Prism height H = 60 cm.
Concept / Approach:Base area A = (1/2) * (product of legs). Then volume V = A * H.
Step-by-Step Solution:
Scale: 5k + 12k + 13k = 30k = 60 ⇒ k = 2Sides: 10, 24, 26 (right triangle)A = (1/2) * 10 * 24 = 120 cm^2V = 120 * 60 = 7200 cm^3Verification / Alternative check:5-12-13 confirms right angle; area via Heron also yields 120 cm^2, validating result.
Why Other Options Are Wrong:6000, 6600, 5400 cm^3 do not match the exact base area times height.
Common Pitfalls:Using hypotenuse in the area formula; mis-scaling side lengths from the ratio; mistaking perimeter for semi-perimeter in Heron’s formula.
Final Answer:7200 cm3