The length L of a rectangle is increasing at a rate of 5 centimeters per second. So we know that DLDT is increasing at 5 centimeters per second while the width DWDT is decreasing or -5 centimeters per second. When length is 5, the width is 12. We want to find the rates of the change of area, the perimeter, the lengths of the diagonal of the rectangle, and determine which of these quantities are increasing or decreasing or constant. So we start with knowing that area is just length times width. So finding the derivative in terms of time, we get DADT equaling DLD T * W + D WD T * L Sticking in the given information, we get DADT equal 35cm ^2 per second. Hence this rate is increasing. The area is getting bigger. We needed to use the product rule there because all three of those variables were changing. And L * W for perimeter, we get perimeter equaling 2 lengths +2 widths. Taking the derivative in terms of time, we get DBTT equaling 2 DLDTS plus 2D WDTS. Sticking that information in, we end up with 0cm per second and hence it's a constant. For the third part, we need to figure out what the rectangle diagonal is going to be. So if we know that the length is five and the width is 12, when we put in that diagonal we have a right triangle. So we know that diagonal is going to equal sqrt 12 ^2 + 5 ^2 for the diagonals just 13. So using the Pythagorean theorem, d ^2 equal l ^2 + W ^2 and taking the derivative in terms of time, we get 2DDD DTS equaling 2LDL DTS plus 2WD WDTS. The twos will all cancel out, so we can get 13 DDDTS equaling 5 * 5 + 12 * -5. The derivative of the diagonal in terms of time is just going to be -35 thirteenths centimeters per second, and because it's a negative, we know it's decreasing or going down.