The volume V of a sphere of radius r changes over time t.

Question

a. Find an equation relating dV/dt to dr/dt.

Accepted Answer

Welcome back, everyone. The surface area A of a cube with side lengths varies over time T. Express DA divided by DT in terms of DS divided by DT. We're given 4 answer choices AS. D A divided by D T equals 24s, DS divided by D T. B says DA divided by D T equals 6Squared multiplied by DS divided by D T. C says DA divided by D T equals 12s multiplied by DS divided by DT and D says DA divided by D T equals 6. S multiplied by DS divided by DT. So we're looking at the surface area of a cube, right? So, essentially, we can just visualize a cube. With a side lengths. Let's go ahead and draw any type of cube. And what we're going to do is recall that a cube is going to have All of those sidelines equal to each other, right? So this is really important to understand. One of those sidelines is S. Now, what is the total surface area? Well, essentially, if we consider the front face, we're going to say that A one, let's call this area. A 1 And essentially we're going to get S squad for the area, right, because it is a square and we have a total of 6 of those faces, right? So A, the total area is going to be 6 multiplied by Squared. So that's the total surface area A of a cube. And now we know that A is a function of time, so we're going to set it. To A of T. and also our side length is a function of time. So what we want to do is differentiate the given equation, applying the derivative with respect to time. On the left side, we're going to get the A divided by D T. On the right hand side, we have our constant, which is 6, and then we have A function that is a composite function, so we need to apply the chain rule. So first of all, we apply the power rule for squad, which gives us 2s, right? That's the derivative of Squad relative to S. and based on the chain rule, we now need to multiply by the derivative of S with respect to time. And now, if we simplify, we're going to get the DA divided by the T equals 12 SDS divided by DT, which essentially corresponds to the answer choice C. Thank you for watching.

The volume V of a sphere of radius r changes over time t.
a. Find an equation relating dV/dt to dr/dt.

Key Concepts

Related Rates

Volume of a Sphere

Chain Rule

Watch next