If the ratio of base radius and height of a cone is 1 : 2 and percentage error in radius is λ %, then the error in its volume is

Question:

If the ratio of base radius and height of a cone is 1 : 2 and percentage error in radius is λ %, then the error in its volume is

(a) λ %
(b) 2 λ %
(c) 3 λ %
(d) none of these

Solution:

(c) 3 λ %

Let the radius of the cone be x, the height be 2x and the volume be y.

$\frac{\Delta x}{x}=\lambda \%$

$\Rightarrow y=\frac{1}{3} \pi x^{2} \times 2 x=\frac{2}{3} \pi x^{3}$

$\Rightarrow \frac{d y}{d x}=2 \pi x^{2}$

$\Rightarrow \frac{\Delta y}{y}=\frac{2 \pi x^{2}}{y} d x=\frac{3}{x} \times \lambda x$

$\Rightarrow \frac{\Delta y}{y}=3 \lambda \%$

Leave a comment