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The volume of a right circular cone is \[4710{\rm{cu}}{\rm{.cm}}\]. If the radius and height of the cone are in the ratio \[3:4\], find its slant height, when \[\pi = 3.14\].

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Last updated date: 16th Jun 2024
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Answer
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Hint: Here we need to find the slant height of the right circular cone. We will first assume the radius and height of the cone to be any variable. We will then write the radius in terms of the height of the cone using the ratio between them. Then we will apply the formula of the cone and then we will substitute the value of the radius in the formula. After simplifying it, we will get the required value of the variable and hence the value of the height and radius of the cone. At last, we will find the value of the slant height of the cone using the formula.

Formula used:
We will use the following formulas:
1. Volume of Cone \[ = \dfrac{1}{3}\pi {r^2}h\], where \[r\] is the radius of the cone and \[h\] is the height of the cone.
2. \[l = \sqrt {{r^2} + {h^2}} \], where \[r\] is the radius of cone, \[l\] is the slant height of the cone and \[h\] is the height of cone.

Complete step by step solution:
Here we need to find the slant height of the right circular cone.
Let the radius of the cone be \[r\] and let the height of the cone be \[h\].
Now we will draw the figure of the cone.
seo images

It is given that the ratio between the radius and height of the cone is \[3:4\].
\[\dfrac{r}{h} = \dfrac{3}{4}\]
We can write it as
\[ \Rightarrow r = \dfrac{3}{4}h\] …………… \[\left( 1 \right)\]
The formula of volume of the cone is equal to \[\dfrac{1}{3}\pi {r^2}h\].
On substituting the value of the radius in the formula, we get
Volume of cone\[ = \dfrac{1}{3}\pi {\left( {\dfrac{3}{4}h} \right)^2}h\]
On further simplifying the terms, we get
\[ \Rightarrow \] Volume of cone \[ = \dfrac{1}{3} \times 3.14 \times \dfrac{9}{{16}} \times {h^3}\]
On multiplying the terms, we get
\[ \Rightarrow \] Volume of cone \[ = 0.58875 \times {h^3}\]
Now, we will substitute the value of the volume of the cone. Therefore, we get
\[ \Rightarrow 4710 = 0.58875 \times {h^3}\]
On dividing both sides by \[0.58875\], we get
\[ \Rightarrow \dfrac{{4710}}{{0.58875}} = {h^3}\]
On further simplification, we get
\[ \Rightarrow 8000 = {h^3}\]
On taking cube root both sides, we get
\[\begin{array}{l} \Rightarrow \sqrt[3]{{8000}} = \sqrt[3]{{{h^3}}}\\ \Rightarrow 20 = h\end{array}\]
The height of the cone is 20 cm.
Now, we will substitute the value of \[h\] in equation \[\left( 1 \right)\].
\[r = \dfrac{3}{4} \times 20 = 15\]
Therefore, the radius of the cone is 15 cm.
Now, we will find the slant height of the cone by using the formula $l=\sqrt{{{r}^{2}}+{{h}^{2}}}$
Substituting the value of height and radius of the cone in the formula, we get
\[l = \sqrt {{{15}^2} + {{20}^2}} \]
Applying the exponent on the terms, we get
\[ \Rightarrow l = \sqrt {225 + 400} \]
Adding the terms, we get
\[ \Rightarrow l = \sqrt {625} = 25\]

Hence, the slant height of the given right circular cone is equal to 25 cm.

Note:
Here we have obtained the length of the slant height of the cone. The slant height of the cone is defined as the shortest distance between the apex of the cone and the base of the cone. A right circular cone is defined as a cone whose axis is the line joining the vertex of the cone to the midpoint of the circular base of the cone. We can make a mistake by taking slant height as the height of the cone and writing the wrong answer.