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

Question

b. At what rate are the lengths of the diagonals of the square changing?

Accepted Answer

Welcome back everyone. The square sides are shrinking at a constant rate of 2 centimeters per second. What is the rate at which the length of the squares diagonal is changing? We're given 4 answer choices A says -2 square root of 2 centimeters per second, B 22 centimeters per second, C-4 centimeters per second, and D-2 centimeters per second. So first of all, let's draw a square, and we have to recall that a square has 4 equal side lengths, right? So let's suppose that we have a side length of A. And now we, if we draw a diagonal, let's call it D, we can simply apply the Pythagorean theorem and solve for D in terms of A. So if we analyze the resultant right triangle, we're going to get a squared. Less a 2 equals d2 so we have applied the Pythagorean theorem, and this gives us 2 a2 equals dred. So D equals square root of 2 a squad, right? Those are positive numbers, so we will only have one solution, and this gives us square root of 2 multiplied by a. And now what we want to do is simply differentiate both sides, right? Because those are changing. D and A are functions of time. So we take DD divided by DT. The rate of change of the diagonal with respect to time is equal to square root of 2, that's our constant multiplied by the derivative of. D A divided by dt, the rate of change of the side length a relative to time. Now what are we given? Well, it says that a square sides are shrinking at a constant rate of 2 centimeters per second. So DA divided by DT is negative because they are shrinking. 2 centimeters per second. Meaning we can now solve for the rate of change of the diagonal with respect to time. We're going to get a square root of 2 multiplied by -2 centimeters per second. Which gives us -2 square root of 2. Centimeter per second and this corresponds to the answer choice a. Thank you for watching.

The sides of a square decrease in length at a rate of 1 m/s.b. At what rate are the lengths of the diagonals of the square changing?

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The sides of a square decrease in length at a rate of 1 m/s.
b. At what rate are the lengths of the diagonals of the square changing?