Geometry

Solids

Medium

A solid cone and a solid cylinder have equal base and equal height. If the radius and height be in the ratio of 12 : 5, the ratio of the total surface area of cone to that of the cylinder is

5 : 7

17 : 25

25 : 7

25 : 34

✅ Correct Option: 4

Let $r=12k$ and $h=5k$ .

The cone needs its slant height, found by Pythagoras:

$l=\sqrt{r^2+h^2}=\sqrt{144k^2+25k^2}=\sqrt{169k^2}=13k$

Solid	Total surface area formula	Value
Cone	$\pi r(r+l)$	$\pi(12k)(12k+13k)=300\pi k^2$
Cylinder	$2\pi r(r+h)$	$2\pi(12k)(12k+5k)=408\pi k^2$

$\text{Ratio}=\dfrac{300\pi k^2}{408\pi k^2}=\dfrac{300}{408}=\dfrac{25}{34}$

So the ratio is $25:34$ .

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