Question

The length of a rectangle is decreasing at the rate of 2 cm/sec, while the width w is increasing at the rate of 3 cm/sec. At what rate are the lengths of the diagonals changing at the instant that l=15 cm and w= 8 cm? Are the diagonals increasing or decreasing in length at this instant? Explain.

I worked through this but am not sure if I did it right or if I needed to work through as much as I did.

A=l*w
dA/dt=l*(dw/dt)+ w*(dl/dt = 45-16= 29cm^2/sec

P=2l*2w =
2(dl/dt)+2(dw/dt)=-4+6=2 cm/sec

Diagonal= (l^2+w^2)^1/2
dD/dt=(2l+2w)(.5)(l^2+w^2)^-.5
=(30+16)(.5)(225+64)^-.5
23/sqrt289=1.35 cm/sec

Did I need to do all this work, or just some of it or did I not even answer the orginally question?

Answer 1

Diagonal= (l^2+w^2)^1/2

then
dD/dt=(2 l dl/dt+2 w dw/dt)(.5)(l^2+w^2)^-.5