Differentiation

Question

A right circular cylindrical metal rod is expanding as it is heated. After t seconds the radius of the rod is (X+1) cm and the length of the rod is (5x+1) cm. The cross-sectional area is increasing at the constant rate of 0.032cm²s-¹.

Find dx/dt when the radius of rod is 2cm ?

Answer 1

Cross section area is:

$A=pi{r}^2$

$dA/dt=pid(r^2)/drdr/dt=pi2rdr/dt$

Substitute (x+1) for r:

$dA/dt=2pi(x+1)dx/dt$

dA/dt = 0.032 when (x=1 = 2:

$0.032=4pidx/dt$

$0.032/(4pi)=dx/dt=0.008/pi$ cm/s