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The length of a rectangle is increasing at a rate of 5 cm/s and its width is increasing at a rate of 4 cm/s. When the length is 9 cm and the width is 6 cm, how fast is the area of the rectangle increasing?

User GLHF
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Answer:

The area of a rectangle is given by the formula A = L x W where A is the area, L is the length, and W is the width.

To find how fast the area of the rectangle is increasing, we need to differentiate the formula for A with respect to time t:

dA/dt = d/dt(L x W)

Using the product rule of differentiation, we get:

dA/dt = dL/dt x W + L x dW/dt

Given that dL/dt = 5 cm/s, dW/dt = 4 cm/s, L = 9 cm, and W = 6 cm, we can substitute these values in the above formula to get the rate at which the area of the rectangle is increasing:

dA/dt = (5 cm/s) x (6 cm) + (9 cm) x (4 cm/s)

dA/dt = 30 cm^2/s + 36 cm^2/s

dA/dt = 66 cm^2/s

Therefore, the area of the rectangle is increasing at a rate of 66 cm^2/s when the length is 9 cm and the width is 6 cm.

User Kyunghoon
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