Question

write a differential formula that estimates the change in the volume $v = \frac{4}{3}\pi r^{3}$ of a sphere when the radius changes from $r_{0}$ to $r_{0}+dr$. choose the correct answer below. a. $dv = \frac{4}{3}\pi r_{0}^{2}dr$ b. $dv = 4\pi r_{0}^{2}dr$ c. $dv = 4\pi r^{2}dr$ d. $dv = 4\pi r_{0}^{3}dr$

Explanation:

Step1: Differentiate the volume formula

Differentiate \( V=\frac{4}{3}\pi r^{3}\) with respect to \(r\). Using the power rule \((x^{n})^\prime = nx^{n - 1}\), we have \(V^\prime=\frac{4}{3}\pi\times3r^{2}=4\pi r^{2}\).

Step2: Use the differential formula

The differential formula is \(dV = V^\prime(r)dr\). When \(r = r_{0}\), \(dV=4\pi r_{0}^{2}dr\).

Answer:

B. \(dV = 4\pi r_{0}^{2}dr\)