Question

In a rectangle, the length is increasing at a rate of 3.02 centimeters per second, while the width is decreasing at a rate of 0.62 centimeters per second. at the time the length is 2 centimeters, and the width is 0.4 centimeters the rate of change of area is

Answer 1

The rate of change of area of the rectangle when the length is 2 cm and the width is 0.4 cm, with the length increasing at 3.02 cm/s and width decreasing at 0.62 cm/s, is -0.032 cm²/s.

Step-by-step explanation:

The student is asking about the rate of change of the area of a rectangle when the length is increasing and the width is decreasing at specific rates. Since the area of a rectangle is calculated by multiplying the length by the width, we can find the rate of change of the area at a specified moment by using these rates and the given dimensions.

To calculate the rate of change of the area (dA/dt) at the moment when the length (l) is 2 centimeters and width (w) is 0.4 centimeters, we use the formula dA/dt = l(dw/dt) + w(dl/dt). The rate of increase of the length (dl/dt) is given as 3.02 cm/s, and the rate of decrease of the width (dw/dt) is given as -0.62 cm/s (negative because it's decreasing). Plugging in the values gives us:

dA/dt = 2 cm * (-0.62 cm/s) + 0.4 cm * (3.02 cm/s)

dA/dt = -1.24 cm²/s + 1.208 cm²/s

dA/dt = -0.032 cm²/s

Therefore, at the time given, the rate of change of area of the rectangle is -0.032 cm²/s, indicating the area is decreasing.