The length of an 18-foot by 15-foot rectangle is increasing at a rate of 5 ft/sec and the width is decreasing at a rate of 3 ft/sec. The length is 18 feet and the width is 15 feet. How fast is the area changing?
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The area of the rectangle is given by A = length × width. Taking the derivative of both sides with respect to time, we get:
dA/dt = (d/dt)(length × width)
Using the product rule for differentiation, we can write this as:
dA/dt = length × (dw/dt) + width × (dl/dt)
where dw/dt represents the rate of change of the width (which is -3 ft/sec) and dl/dt represents the rate of change of the length (which is 5 ft/sec).
Plugging in the given values, we have:
dA/dt = 18 × (-3) + 15 × 5
= -54 + 75
= 21
Therefore, the area of the rectangle is changing at a rate of 21 square feet per second.
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