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Curved surface area of a cone is $308{\text{ c}}{{\text{m}}^2}$ and its slant height is $14{\text{ cm}}$. Find
${\text{(i)}}$ radius of the base
${\text{(ii)}}$ total surface area of the cone.

Answer Verified Verified
Hint- Here, we will be using the formulas for curved surface area and total surface area of the cone.

Given, Curved surface area of a cone is ${{\text{A}}_{\text{c}}} = 308{\text{ c}}{{\text{m}}^2}$ and its slant height is $l = 14{\text{ cm}}$
${\text{(i)}}$ Since, the formula for the curved surface of a cone having radius of the base as $r$ and slant height as $l$ is given by ${{\text{A}}_{\text{c}}} = \pi rl$
Using above formula, we can write
${{\text{A}}_{\text{c}}} = \pi rl \Rightarrow 305 = \dfrac{{22}}{7} \times r \times 14 \Rightarrow r = \dfrac{7}{{22 \times 14}} \times 305 \Rightarrow r = 6.93{\text{ cm}}$
Therefore, the radius of the base of the given cone is 6.93 cm.
${\text{(ii)}}$ Also, we know that the formula for the total surface of a cone having radius of the base as $r$ and slant height as $l$ is given by ${{\text{A}}_{\text{s}}} = \pi {r^2} + \pi rl = \pi r\left( {r + l} \right)$
Using the above formula, we get
${{\text{A}}_{\text{s}}} = \pi r\left( {r + l} \right) = \dfrac{{22}}{7} \times 6.93 \times \left( {6.93 + 14} \right) = \dfrac{{22}}{7} \times 6.93 \times 20.93 = 455.85{\text{ c}}{{\text{m}}^2}$
Therefore, the total surface area of the given cone is $455.85{\text{ c}}{{\text{m}}^2}$.

Note- In these types of problems, we have to make sure that all the units of given values are the same. Also, the total surface area of the cone is the sum of its curved surface area (i.e., $\pi rl$) and the area of the base (i.e., $\pi {r^2}$).
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