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The radius and height of the cone are in the ratio of 4:3. The area of the base is 154 square centimeter. Find the curved surface area of the cone.

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: Assume the radius (r) of the cone as 4x and its height as (h) as 3x. Now, use the formula for the area of the circular base of the cone as $\pi {{r}^{2}}$ and substitute it with 154 to solve for the value of x. Once the value of x is found, substitute it in the assumed value of h to find its value. Finally, apply the formula Curved surface area of cone = $\pi r\sqrt{{{r}^{2}}+{{h}^{2}}}$ and substitute the obtained values to get the answer. Use the value $\pi =\dfrac{22}{7}$.

Complete step by step answer:
Here we have been provided with the ratio of radius and height of a cone along with its area of base. We have been asked to find the curved surface area of the cone. First we need to find the value of radius and height.
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Now, the ratio of radius and height is 4:3 so we can assume the radius as 4x and the height as 3x. The area of the base is 154 square centimeter. We know that the base of the cone is circular in nature so applying the formula for the area of a circle we get,
$\Rightarrow \pi {{r}^{2}}=154$
Substituting the values r = 4x and $\pi =\dfrac{22}{7}$ we get,
$\begin{align}
  & \Rightarrow \dfrac{22}{7}\times {{\left( 4x \right)}^{2}}=154 \\
 & \Rightarrow {{\left( 4x \right)}^{2}}=7\times 7 \\
 & \Rightarrow {{\left( 4x \right)}^{2}}={{7}^{2}} \\
\end{align}$
Taking square root both the sides we get,
$\begin{align}
  & \Rightarrow x=7 \\
 & \Rightarrow x=\dfrac{7}{4} \\
\end{align}$
Substituting the value of x in assumed values of r and h we get r = 7 cm and h = $\dfrac{21}{4}$ cm. we know that that the curved surface area of a cone is given as $\pi r\sqrt{{{r}^{2}}+{{h}^{2}}}$, so substituting the obtained values we get,
$\Rightarrow $ Curved surface area = $\dfrac{22}{7}\times 7\times \sqrt{{{\left( 7 \right)}^{2}}+{{\left( \dfrac{21}{4} \right)}^{2}}}$
On simplifying we get,
$\Rightarrow $ Curved surface area = 192.5 $c{{m}^{2}}$
Hence, the curved surface area of the cone is 192.5 square centimeter.

Note: One important thing you may note is that in the formula for the curved surface area of the cone which is $\pi r\sqrt{{{r}^{2}}+{{h}^{2}}}$ we have the relation $\sqrt{{{r}^{2}}+{{h}^{2}}}=l$. Here ‘l’ is called the slant height of the cone, so you can also remember the formula as $\pi rl$ for the curved surface area. The total surface area of a cone is $\pi r\left( l+r \right)$. Note that we have used the value $\pi =\dfrac{22}{7}$ for making our calculations easy.
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