Question

The radius and height of a right cone are in the ratio $3:4$. The ratio of total surface area to curved surface area of the cone is

• A. $2:3$
• B. $3:5$
• C. $7:9$
• D. $8:5$

Answer 1

The correct option is D

$8:5$

Given: $r:h=3:4$ Let $r=3x$ and $h=4x$

$\Rightarrow$ In the above figure, $BC=r=3x$ and $h=AB=4x$

In $△ABC$

$A{B}^2+B{C}^2=A{C}^2$

$\Rightarrow 16x^2+9x^2=A{C}^2$

$\Rightarrow AC=\sqrt{25x^2}=5x$

$\Rightarrow$ Slant height of the cone is $l=5x$

Required ratio$=\frac{\text{total surface area}}{\text{curved surface area}}=\frac{\pi r(r+l)}{\pi rl}$

$=\frac{\pi \times 3x(3x+5x)}{\pi \times 3x\times 5x}$

$=8:5$