Cone scaled in radius and height — new volume ratio: The base radius and height of a cone are both doubled. What is the ratio of the new volume to the original volume?

Introduction / Context:Volume of a cone is V = (1/3)πr^2h. Scaling r and h scales volume by r^2h, i.e., by the square of radius scale times the height scale.

Given Data / Assumptions:r → 2r and h → 2h.

Concept / Approach:New/old volume factor = (2^2) * (2) = 8.

Step-by-Step Solution:V_new / V_old = (r_new^2 h_new)/(r^2 h) = (4r^2 * 2h)/(r^2 h) = 8

Verification / Alternative check:Pick numbers (e.g., r=1, h=1) to see old V=(1/3)π, new V=(8/3)π → ratio 8:1.

Why Other Options Are Wrong:Other ratios ignore the quadratic dependence on radius and the linear dependence on height.

Common Pitfalls:Adding scale factors instead of multiplying; forgetting radius is squared in volume.

Final Answer:8 : 1