Question

6) 7 marks A spherical balloon is being inflated. Using limit theory only: a) Find the instantaneous rate of change of its surface area (SA = 4nr2) with respect to its radius r when r = 10cm b) At another time, the instantaneous rate of change is 481 cm2/cm. What is the Surface Area of the balloon at this time?

Answer 1

a)

$\small SA = 4\pi r^2$

$\small \frac{\partial }{\partial r}SA = 8\pi r$

at r = 10 cm. so,

$\small \frac{\partial }{\partial r}SA = 8\pi \times 10$

$\small \frac{\partial }{\partial r}SA = 80\pi$

b)

given

$\small \frac{\partial }{\partial r}SA = 48\pi$

$\small 4\pi r = 48\pi$

$\small r = 12 \ cm$

so, surface area

$\small SA = 4\pi r^2$

$\small SA = 4\pi (12)^2$

$\small SA = 576 \pi \ cm^2$