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Chapter 2: Problem 87

The volume of a cube with sides of length \(s\) is given by \(V=s^{3} .\) Find the rate of change of the volume with respect to \(s\) when \(s=4\) centimeters.

Short Answer

Expert verified
The rate of change of the volume with respect to the length of the side of the cube when \(s=4\) centimeters is 48 cubic centimeters per centimeter.

Step by step solution

01

Express the Problem Mathematically

The volume \(V\) of a cube with side length \(s\) is given by the formula \(V=s^{3} .\) The rate of change of the volume with respect to \(s\) is given by the derivative \(\frac{dV}{ds}\). To find this derivative when \(s=4\), we will plug \(s=4\) into the derivative equation after finding it.
02

Differentiate the Volume Equation

Taking the derivative of both sides of the volume equation with respect to \(s\), we obtain: \(\frac{dV}{ds} = 3s^{2}\).
03

Substitute the Given Value of \(s\)

Substitution \(s=4\), from the given problem, into the derivative equation results in: \(\frac{dV}{ds} = 3(4)^{2} = 48\).

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