Question

The radius of the base of a cone is increasing at the rate of 3 cm/min and the altitude is decreasing at the rate of 4 cm/min. The rate of change of lateral surface when the radius is 7 cm and altitude is 24 cm is

• A. $54\pi$ cm${}^2$/min.
• B. $7\pi$ cm${}^2$/min.
• C. $27\pi$cm${}^2$/min.
• D. none of these

Answer 1

Answer: (A)

$54\pi$ cm${}^2$/min.

Slant height $l^2=r^2+h^2$
$\Rightarrow l=25cm$
Also,$2l\frac{dl}{dt}=2r\frac{dr}{dt}+2h\frac{dh}{dt}$
$\Rightarrow \frac{dl}{dt}=-3cm/min$
Surface Area of cone$S=\pi rl$
$\Rightarrow \frac{dS}{dt}=\pi (r\frac{dl}{dt}+l\frac{dr}{dt})$
$\Rightarrow \frac{dS}{dt}=54\pi c{m}^2/min$