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.

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