Question

the length of a rectangle is increasing at the rate of 7 cm/s, and its width is increasing at a rate of 2 cm/s. when the length is 30 cm and the width is 20 cm, how fast is the area of the rectangle increasing?

Answer 1

The area of the rectangle is increasing at a rate of 200 cm²/s when the length is 30 cm and the width is 20 cm.

Step-by-step explanation:

The question is about how fast the area of a rectangle is increasing given that the dimensions of the rectangle (length and width) are increasing at constant rates. To solve this problem, we can use related rates, which are a part of calculus. Given that the length (L) is increasing at 7 cm/s and the width (W) is increasing at 2 cm/s, we can set up the formulas related to the area (A) of a rectangle, which is A = L * W.

Now, to find the rate at which the area of the rectangle is increasing, we will take the derivative of both sides of the equation A = L * W concerning time (t). This yields dA/dt = L * (dW/dt) + W * (dL/dt). Substituting the given lengths and rates into the equation gives us dA/dt = 30 cm * 2 cm/s + 20 cm * 7 cm/s = 60 cm²/s + 140 cm²/s = 200 cm²/s. Therefore, the area of the rectangle is increasing at a rate of 200 cm²/s when the length is 30 cm and the width is 20 cm.