Question
A right circular cylinder and aright 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 cylinder and a cone respectively
$\therefore$ Slant height of cone $=\sqrt{\text{r}^2+\text{h}^2}$
Now curved surface area of cylinder $=2\pi\text{rh}$
and cureved surface area of cone $=\pi\text{rl}$
$=\pi\text{r}\sqrt{\text{r}^2+\text{h}^2}$
$\because$ their ratio is $8 : 5$
$\therefore\frac{2\pi\text{rh}}{\pi\text{r}\sqrt{r^2+\text{h}^2}}=\frac{8}{5}\Rightarrow\frac{2\text{h}}{\sqrt{r^2+\text{h}^2}}=\frac{8}{5}$
$\Rightarrow\frac{4\text{h}^2}{\text{r}^2+\text{h}^2}=\frac{64}{25}$ (squaring both side)
$\Rightarrow\frac{\text{h}^2}{\text{r}^2+\text{h}^2}=\frac{16}{25}$ (Dividing by 4)
$\Rightarrow25\text{h}^2=16\text{r}^2+16\text{h}^2$
$\Rightarrow25\text{h}^2-16\text{h}^2=16\text{r}^2\Rightarrow9\text{h}^2=16\text{r}^2$
$\Rightarrow\frac{\text{r}^2}{\text{h}^2}=\frac{9}{16}\Rightarrow\Big(\frac{\text{r}}{\text{h}}\Big)^2=\Big(\frac{3}{4}\Big)^2$
$\therefore\frac{\text{r}}{\text{h}}=\frac{3}{4}\Rightarrow\text{r}:\text{h}=3:4$
$\therefore$ Ratio of r and $h = 3 : 4$
$\therefore$ Slant height of cone $=\sqrt{\text{r}^2+\text{h}^2}$
Now curved surface area of cylinder $=2\pi\text{rh}$
and cureved surface area of cone $=\pi\text{rl}$
$=\pi\text{r}\sqrt{\text{r}^2+\text{h}^2}$
$\because$ their ratio is $8 : 5$
$\therefore\frac{2\pi\text{rh}}{\pi\text{r}\sqrt{r^2+\text{h}^2}}=\frac{8}{5}\Rightarrow\frac{2\text{h}}{\sqrt{r^2+\text{h}^2}}=\frac{8}{5}$
$\Rightarrow\frac{4\text{h}^2}{\text{r}^2+\text{h}^2}=\frac{64}{25}$ (squaring both side)
$\Rightarrow\frac{\text{h}^2}{\text{r}^2+\text{h}^2}=\frac{16}{25}$ (Dividing by 4)
$\Rightarrow25\text{h}^2=16\text{r}^2+16\text{h}^2$
$\Rightarrow25\text{h}^2-16\text{h}^2=16\text{r}^2\Rightarrow9\text{h}^2=16\text{r}^2$
$\Rightarrow\frac{\text{r}^2}{\text{h}^2}=\frac{9}{16}\Rightarrow\Big(\frac{\text{r}}{\text{h}}\Big)^2=\Big(\frac{3}{4}\Big)^2$
$\therefore\frac{\text{r}}{\text{h}}=\frac{3}{4}\Rightarrow\text{r}:\text{h}=3:4$
$\therefore$ Ratio of r and $h = 3 : 4$
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