Mathematics

Mathematics

Problem

4. An edge of a variable cube is increasing at the rate of 3 cm/s3 \mathrm{~cm} / \mathrm{s}. How fast is the volume of the cube increasing when the edge is 10 cm long?
Intuitive explanation

This is a related rates problem where you connect how one quantity changes to how another changes. The key insight is to write the volume formula V=s3V = s^3, then differentiate both sides with respect to time to relate dVdt\frac{dV}{dt} (what you want) to dsdt\frac{ds}{dt} (what you're given). Once you have that relationship, substitute the specific edge length to find the answer.

Intuitive explanation

This is a related rates problem where you connect how one quantity changes to how another changes. The key insight is to write the volume formula V=s3V = s^3, then differentiate both sides with respect to time to relate dVdt\frac{dV}{dt} (what you want) to dsdt\frac{ds}{dt} (what you're given). Once you have that relationship, substitute the specific edge length to find the answer.

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