Question
A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surfaces are in the ratio 8 : 5, determine the ratio of the radius of the base to the height of either of them.
✓
Answer
Let r and h be the radius and height of a circular cylinder and also of a cone, then curved
surface area of the cylinder $=2\pi\text{rh}$
and curved surface area of cone
$=\pi\text{rl}=\pi\text{r}\sqrt{\text{h}^2+\text{r}^2}$
but they are in the ratio 8 : 5
$\frac{2\pi\text{rh}}{\pi\text{r}\sqrt{\text{h}^2+\text{r}^2}}=\frac{8}{5}\Rightarrow\frac{2\text{h}}{\sqrt{\text{h}^2+\text{r}^2}}=\frac{8}{5}$
$\frac{4\text{h}^2}{\text{h}^2+\text{r}^2}=\frac{64}{25}$ (squaring on both sides)
$\frac{\text{h}_2}{\text{h}^2+\text{r}^2}=\frac{16}{25}$
$\Rightarrow25\text{h}^2=16\text{h}^2+16\text{r}^2$
$\Rightarrow25\text{h}^2-16\text{h}^2=16\text{r}^2$
$\Rightarrow9\text{h}^2=16\text{r}^2$
$\Rightarrow\frac{\text{r}^2}{\text{h}^2}=\frac{9}{16}=\Big(\frac{3}{4}\Big)^2$
$\Rightarrow\frac{\text{r}}{\text{h}}=\frac{3}{4}$
$\therefore$ Ratio of r and h = 3 : 4
Hence ratio of radius and height = 3 : 4.
surface area of the cylinder $=2\pi\text{rh}$
and curved surface area of cone
$=\pi\text{rl}=\pi\text{r}\sqrt{\text{h}^2+\text{r}^2}$
but they are in the ratio 8 : 5
$\frac{2\pi\text{rh}}{\pi\text{r}\sqrt{\text{h}^2+\text{r}^2}}=\frac{8}{5}\Rightarrow\frac{2\text{h}}{\sqrt{\text{h}^2+\text{r}^2}}=\frac{8}{5}$
$\frac{4\text{h}^2}{\text{h}^2+\text{r}^2}=\frac{64}{25}$ (squaring on both sides)
$\frac{\text{h}_2}{\text{h}^2+\text{r}^2}=\frac{16}{25}$
$\Rightarrow25\text{h}^2=16\text{h}^2+16\text{r}^2$
$\Rightarrow25\text{h}^2-16\text{h}^2=16\text{r}^2$
$\Rightarrow9\text{h}^2=16\text{r}^2$
$\Rightarrow\frac{\text{r}^2}{\text{h}^2}=\frac{9}{16}=\Big(\frac{3}{4}\Big)^2$
$\Rightarrow\frac{\text{r}}{\text{h}}=\frac{3}{4}$
$\therefore$ Ratio of r and h = 3 : 4
Hence ratio of radius and height = 3 : 4.
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