If the original 24 m edge length x of a cube decreases at the ...

Question

If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’s

a. surface area change?

Accepted Answer

Welcome back, everyone. A circular pond's radius is shrinking due to evaporation at a rate of 1 m per hour. When the radius is 5 m, at what rate is the pond's surface area decreasing? A 5 square per hour, B 50 squad per hour. C 20 square per hour, and D 10 squad per hour. First of all, we know that. The pond is modeled in terms of a circle, and we have to recall that the area of a circle is by R squad where R is radius. We want to identify the rate of change of ponds surface areas. We want to calculate DA divided by DT, the rate of change of area with respect to time, right? Because A is a function of R and R is a function of time. We can recognize that A is a composite function A of R of T, so when we Differentiate A, we have to apply the chain rule. We have to differentiate A with respect to R and multiply that by the derivative of R with respect to time. So let's differentiate a. With respect to radius, we have pi, which is a constant multiplied by the derivative of R squared, which gives us pi multiplied by 2 R, and that's 2 R. And then we can show that the derivative of A with respect to time is 2 pi R multiplied by DR divided by DT. Now let's consider what's given in the problem. We know that the radius is shrinking at a rate of 1 m per hour. So this means that the R divided by the T is equal to -1 m per hour. It's shrinking, so the rate of change is negative. And we also know that the radius is 5 m at. The point of interest. So we want to evaluate the A divided by DT. When DR divided by DT is equal to -1 and R equals 25. So we basically have to substitute those into the equation to pi R DR divided by DT, which gives us 2 pi multiplied by 5. Meters multiplied by -1 m per hour. And this gives us Negative sum pi meter square per hour. Notice that the rate of change of area with respect to time is negative, meaning it is decreasing. We don't need to include the negative sign in the answer because decreasing. In the context of the problem already implies that there's going to be a negative sign, right? So the answer to this problem corresponds to the answer to the. The pond's surface area is decreasing at a rate of 10 m per hour. Thank you for watching.

If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’s
a. surface area change?

Key Concepts

Related Rates

Surface Area of a Cube

Chain Rule in Differentiation

Watch next