In Exercises 5-8, assume that the radius r of a sphere is...

In Exercises $5-8,$ assume that the radius $r$ of a sphere is expanding at a rate of 30 $\mathrm{cm} / \mathrm{min}$ . The volume of a sphere is $V=\frac{4}{3} \pi r^{3}$ and its surface area is 4$\pi r^{2}$ . Determine the given rate. Surface area with respect to time when $r$=40 cm.

Key Concepts

Surface Area of a Sphere

The surface area of a sphere is given by the formula 4?r², where r is the radius. Differentiating this formula with respect to the radius gives an expression that, when multiplied by the rate of change of the radius, provides the rate of change of the surface area with respect to time.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. In the context of related rates, it allows us to differentiate a function with respect to one variable while accounting for the fact that the variable itself is a function of time, thus linking the derivative of the function to the derivative of the variable.

Related Rates

Related rates problems involve finding the rate at which one quantity is changing by relating it to another quantity whose rate of change is known. This typically requires differentiating an equation that relates the different quantities with respect to time, thereby connecting their instantaneous rates of change.

Transcript

0:00 Hi there.

00:02 In this problem we are asked to find the rate of change of surface area with respect to time.

00:10 In other words, we are looking for, let's use a for area, and so we're looking for d -a -d -t in the end.

00:24 That's what we want to figure out.

00:29 What else do we know? well, we are told in these problems that the radius is expanding at 30 centimeters per minute.

00:40 So what that means is that drdt, the rate of change of the radius, is positive 30 centimeters per minute.

00:53 And do we know anything else? well, we do know that we care about the rate of change of surface area specifically when r equals 40.

01:00 So we're going to care about the moment when r equals 40.

01:07 So that's all the information we have.

01:10 Now to solve this, since the question is about surface area, we're going to need the equation or the formula for the surface area of a sphere which as the problem gives us is 4 times pi times the radius squared and since we care about rates we need to take the derivative with respect to t so let's do that on both sides and see what we get so on the left side the derivative of a with respect to t is a definition d a d t on the right on the 4 and the pi are constants, the derivative of r squared is 2r.

01:57 Then we need to multiply, since r is a function of time, you multiply by dr d t.

02:05 In other words, we get 8 pi r, d, d, t...

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