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11 April, 04:10

The length l of a rectangle is decreasing at a rate of 1 cm/sec, while its width w is increasing at the rate of 3 cm/sec. Find the rates of change of the perimeter and the length of one diagonal at the instant when l = 15 cm and w = 6 cm.

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  1. 11 April, 05:43
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    when l = 15 cm and w = 6 cm the perimeter will change at a rate P = 4 cm/s

    and the diagonal at a rate C = 0.185 cm/s

    Step-by-step explanation:

    representing the rate of change of width w as W=dw/dt = 3 cm/s (t=time) and the rate of change of length l as L=dl/dt = - 1 cm/sec

    then the perimeter p will be

    p = 2*l + 2*w

    the rate of change of p will be P=dp/dt

    P=dp/dt = 2*dw/dt + 2*dw/dt = 2*W + 2*L

    P = 2*W + 2*L

    replacing values

    P = 2*W + 2*L = 2*3 cm/s + 2 * (-1 cm/s) = 4 cm/s

    P = 4 cm/s

    for the diagonal c

    c = √ (l² + w²)

    the rate of change of c will be C=dc/dt

    C=dc/dt = 1 / (2*√ (l² + w²)) * (2*l*dl/dt + 2*w*dw/dt) = (l*L+w*W) / √ (l² + w²)

    when l = 15 cm and w = 6 cm

    C = (l*L+w*W) / √ (l² + w²) = (15 cm * (-1 cm/sec) + 6 cm*3 cm/s) / √[ (15 cm) ² + (6 cm) ²] = 1/√29 = 0.185 cm/s

    C = 0.185 cm/s
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