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A right circular cylinder and a right circular cone have equal base 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.

Last updated date: 20th Jun 2024
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Hint: Start by assuming the dimensions of the given shapes as some variable and use the formula for curved surface area as per the object. Use the data of ratios of areas given in the question, Substitute all the values and simplify, we will get the desired ratio.

Given,
Ratio of curved surface areas of cylinder and cone = 8:5
Radius of right cylinder = Radius of right circular cone = r (assume)
Also,
Height of cylinder = Height of right circular cone = h (assume)
Now , we know Curved surface area of right circular cylinder = $2\pi rh$
And , Curved surface area of right circular cone = $\pi rl$, where $l$ is the slant height of cone and it can be found by using the formula $l = \sqrt {{h^2} + {r^2}}$
So , Curved surface area of circular cone can be written as = $\pi r\sqrt {{h^2} + {r^2}}$
$\dfrac{{{\text{Curved surface area of cylinder }}}}{{{\text{Curved surface area of cone }}}} = \dfrac{8}{5}$
Substituting the formulas , we get

$\dfrac{{2\pi rh{\text{ }}}}{{\pi r\sqrt {{h^2} + {r^2}} {\text{ }}}} = \dfrac{8}{5} \\ \Rightarrow \dfrac{{2h}}{{\sqrt {{h^2} + {r^2}} {\text{ }}}} = \dfrac{8}{5} \\$
Squaring both the sides , we get
$\dfrac{{4{h^2}}}{{{h^2} + {r^2}{\text{ }}}} = \dfrac{{64}}{{25}}$
Dividing the numerator and denominator in L.H.S by ${h^2}$, we get
$\dfrac{4}{{1 + \dfrac{{{r^2}}}{{{h^2}}}{\text{ }}}} = \dfrac{{64}}{{25}}$
By cross multiplication , we get
$100 = 64(1 + \dfrac{{{r^2}}}{{{h^2}}}) \\ \Rightarrow \dfrac{{100}}{{64}} = 1 + \dfrac{{{r^2}}}{{{h^2}}} \\$
Now shifting 1 to the other side of the equation. We get,
$\dfrac{{100}}{{64}} - 1 = \dfrac{{{r^2}}}{{{h^2}}} \\ \dfrac{{36}}{{64}} = \dfrac{{{r^2}}}{{{h^2}}} \\$
Taking square roots both the sides , we get
$\sqrt {\dfrac{{36}}{{64}}} = \sqrt {\dfrac{{{r^2}}}{{{h^2}}}} \\ \Rightarrow \pm \dfrac{6}{8} = \dfrac{r}{h} \\$
Which on further simplification , give us
$\pm \dfrac{3}{4} = \dfrac{r}{h}$
Since r and h are the distances or lengths , they can never be negative .
Therefore, the ratio of the radius of the base to the height is $\dfrac{r}{h} = \dfrac{3}{4}$.

Note: Similar questions can be asked with different geometrical shapes and objects, Follow the same procedure as above to get the desired ratio as per mentioned in the question. Attention must be given while substituting the values and any negative values obtained in ratios is to be neglected as even if you take two negative numbers , the ratio would still be positive.