The length I of a rectangle is decreasing at a rate of 7 cm/sec while the width

Question

The length I of a rectangle is decreasing at a rate of 7 cm/sec while the width w is increasing at a rate of 7cm/sec. When l = 12 cm and w = 5 cm, find the rates of change of the area, the perimeter, and the lengths of the diagonals of the rectangle. Determine which of these quantities are increasing, decreasing, or constant. The rate of change of the area of the rectangle is 49 cm2/sec. Is the area increasing, decreasing, or constant? Constant Increasing Decreasing The rate of change of the perimeter of the rectangle is 0 cm/sec. Is the perimeter increasing, decreasing, or constant? Increasing Decreasing Constant

Explanation / Answer

L = 12

dL/dt = -7

W = 5

dW/dt = 7

A = LW

dA/dt = (dL/dt)W + L(dW/dt)

dA/dt = (-7)(5) + (12)(7) = 49

Recall that a function is increasing when its derivative is positive.

Since dA/dt = 49 > 0, the area of the rectangle is increasing.

P = 2L + 2W = 2(L + W)

dP/dt = 2(dL/dt + dW/dt)

dP/dt = 2(-7 + 7) = 0

Recall that a function is constant when its derivative is zero.

Since dP/dt = 0, the perimeter of the rectangle is constant.

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