Shrinking square The sides of a square decrease in length at a...

Question

Shrinking square The sides of a square decrease in length at a rate of 1 m/s.

a. At what rate is the area of the square changing when the sides are 5 m long?

Accepted Answer

Welcome back everyone to another video. The side length of a square is shrinking at a rate of 0.5 m per second. Determine the rate of change of the area when the side length is 10 m. We're given four answer choices A says -5 m squad per second, B 10 m square per second, C-20 meter squad per second, and D-50 meters squad per second. So now, let's visualize a square and let's suppose that. The square has a side length equal to a. So essentially every side measures A, specifically a meters, right? And we have to recall that the area of a square will be equal to a squad. So now, let's understand that area in this case is a function of time, and our side length, well, relative to time, it is also changing, right? It says that it is shrinking at a specific rate. So lowercase a, which is our side length, is also a function of time. So what we want to do is perform implicit differentiation on the given equation. Now, differentiating relative to time we're going to get. The derivative of area with respect to time is equal to the derivative of a squared, and essentially first of all we have to differentiate a squared relative to a before we move on to time. And this means that we're applying the power rule and the derivative of a squared is to a, but A is a function of time. So we applied the chain rule and we get multiplied by the lowercase a divided by DT, right? So we want to use this equation and we want to find the given variables. So the side length is shrinking at a rate of 0.5 m per second. And since lowercase a is our side length, we can say that d a divided by d T is equal to negative. Now why negative? Well, it says shrinking, right? So it's decreasing. And we have 0.5 m per second. We also know that the side length at the point of interest, lowercase a is equal to 10 m. So we can easily solve for the rate of change of area D A divided by DT, which is going to be equal to 2 multiplied by a centimeters multiplied by the d a divided by dt, which is 0.5 m per second. And now simplifying, we're going to get the rate of change of area with respect to time is equal to 20 m multiplied by -0.5 m per second, right? So that's -10. Meters squad per second which corresponds to the choice b. Thank you for watching.

Shrinking square The sides of a square decrease in length at a rate of 1 m/s.
a. At what rate is the area of the square changing when the sides are 5 m long?

Key Concepts

Related Rates

Area of a Square

Chain Rule

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