write a differential formula that estimates the change in the volume $v = \\frac{4}{3}\\pi r^{3}$ of a…

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. $dv = 4\\pi r_{0}^{3}dr$

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. $dv = 4\\pi r_{0}^{3}dr$

Answer

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)