Cone volume change — height tripled, radius halved: If the height of a right circular cone is increased by 200% (i.e., tripled) and the base radius is reduced by 50%, what is the net change in the cone’s volume?

Introduction / Context:
Volume V of a cone is (1/3)πr^2h. Scale factors multiply: radius affects volume quadratically; height affects it linearly.

Given Data / Assumptions:
h_new = 3h (200% increase), r_new = 0.5r (50% decrease).

Concept / Approach:
Compute factor f = (r_new^2 h_new)/(r^2 h) = (0.5^2)*3 = 0.25*3 = 0.75.

Step-by-Step Solution:
V_new = 0.75 V_oldChange = −25% (a decrease of one quarter)

Verification / Alternative check:
Pick r=2, h=2 ⇒ V_old = (1/3)π*4*2 = (8/3)π; new r=1, h=6 ⇒ V_new = (1/3)π*1*6 = 2π; 2π/(8/3)π = 0.75.

Why Other Options Are Wrong:
They do not align with the combined quadratic and linear scaling.

Common Pitfalls:
Adding the percentages instead of multiplying the factors.

Final Answer:
decreases by 25%