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Question

A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio $$8:5$$, show that the radius of each is to the height of each as $$3:4$$.

Solution
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For cylinder, we have
base radius$$=$$r, height$$=$$h

$$\therefore S_1=$$Curved surface$$=2\pi rh$$

For cone, we have
$$l=\sqrt{r^2+h^2}$$ and, $$S_2=\pi rl=\pi r\sqrt{r^2+h^2}$$

We have,
$$\dfrac{S_1}{S_2}=\dfrac{8}{5}$$

$$\Rightarrow \dfrac{2\pi rh}{\pi r\sqrt{r^2+h^2}}=\dfrac{8}{5}$$

$$\Rightarrow \dfrac{2h}{\sqrt{r^2+h^2}}=\dfrac{8}{5}$$

$$\Rightarrow \dfrac{4h^2}{r^2+h^2}=\dfrac{64}{25}$$

$$\Rightarrow 25h^2=16r^2+16h^2\Rightarrow 9h^2=16r^2\Rightarrow 3h=4r$$

$$\Rightarrow \dfrac{r}{h}=\dfrac{3}{4}$$.

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