1 Answer

Teiem · Answer 1 · 2023-02-10T18:31:14+0000

The formula for the volume of a sphere is given by:

$V=(4)/(3)\pi r^3$

Since we are asked the rate of change of the volume with repect to time, we take the derivative on both sides, taking into account the chain rule:

$(dV)/(dt)=(d((4)/(3)\pi r^3))/(dt)$

taking out the constants:

$(dV)/(dt)=(4)/(3)\pi(d(r^3))/(dx)$

Now we derivate, using the chain rule, that is:

$(df(g(x)))/(dx)=f^(\prime)(x)g^(\prime)(x)$

Applying the rule:

$(dV)/(dt)=(4)/(3)\pi(3r^2)(dr)/(dt)$

Simplifying:

$(dV)/(dt)=4\pi(r^2)(dr)/(dt)$

We have the following known values:

$\begin{gathered} (dr)/(dt)=\frac{2\operatorname{cm}}{s} \\ r=3\operatorname{cm} \end{gathered}$

Replacing we get:

$(dV)/(dt)=4\pi(3)^2(2)$

Solving we get:

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