The length of a rectangle is increasing at a rate of 3 cm/s and its width is increasing at a rate of 9 cm/s. When the length is 15 cm and the width is 10 cm, how fast is the area of the rectangle increasing? cm2/s
Question
The length of a rectangle is increasing at a rate of 3 cm/s and its width is increasing at a rate of 9 cm/s. When the length is 15 cm and the width is 10 cm, how fast is the area of the rectangle increasing? cm2/s
Solution
The area of a rectangle is given by the formula A = lw, where l is the length and w is the width.
Given that the length l is increasing at a rate of 3 cm/s (dl/dt = 3 cm/s) and the width w is increasing at a rate of 9 cm/s (dw/dt = 9 cm/s), we want to find how fast the area A is increasing when l = 15 cm and w = 10 cm.
We can find this by taking the derivative of the area with respect to time t, which gives us dA/dt = l(dw/dt) + w(dl/dt).
Substituting the given values into this equation gives us dA/dt = 15(9) + 10(3) = 135 + 30 = 165 cm²/s.
So, the area of the rectangle is increasing at a rate of 165 cm²/s.
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