Question

In Exercises \(35-40,\) write a differential formula that estimates the given change in volume or surface area. Then use the formula to estimate the change when the dependent variable changes from 10 \(\mathrm{cm}\) to 10.05 \(\mathrm{cm} .\) Volume The change in the volume \(V=(4 / 3) \pi r^{3}\) of a sphere when the radius changes from \(a\) to \(a+d r\)

Short answer

The estimated change in volume is approximately \(20.94 \, \text{cm}^{3}\).

Step-by-step solution

Differentiate the Volume Formula

Differentiate the given volume formula \(V=(4 / 3) \pi r^{3}\) with respect to 'r'. The derivative of \(r^{3}\) is \(3r^{2}\), so \(dV / dr = 4\pi r^{2}\).

Find the Differential dV

The differential of V is then found by multiplying the derivative \(dV / dr\) by \(dr\), the change in radius. So, \(dV = (4\pi r^{2}) dr\).

Plug in the Values and Calculate

We're given that the radius changes from 10cm to 10.05cm, so \(dr\) is 0.05. Let's plug r=10cm and \(dr\) = 0.05cm into our differential formula \(dV = (4\pi r^{2}) dr\). Calculation will give the estimated change in volume.