Question

The total surface area of a conical tent is 920 square meters and its radius 14 m. Find its slant height (Round off your answer to the nearest whole number).

Answer 1

Hint:
Here, we need to find the slant height of the conical tent. We will use the formula of the total surface area to find the slant height of a cone. The slant height is the distance from the lateral side of the base to the apex of the cone.
Formula Used: The total surface area of a cone is given by the formula \[T.S.A. = \pi r\left( {l + r} \right)\], where \[l\] is the slant height of the cone, and \[r\] is the radius of the cone.

Complete step by step solution:
We will use the formula for total surface area of a cone to get the slant height.
The total surface area of a cone is given by the formula \[T.S.A. = \pi r\left( {l + r} \right)\], where \[l\] is the slant height of the cone, and \[r\] is the radius of the cone.
We can observe that the formula does not require the height of the cone.
Thus, we do not need to find the height of the cone.
Substituting \[T.S.A. = 920{\rm{ }}{{\rm{m}}^2}\] and \[r = 14{\rm{ m}}\] in the formula for total surface area of the cone, we get
\[920 = \pi \left( {14} \right)\left( {l + 14} \right)\]
Rewriting the equation, we get
\[ \Rightarrow 920 = 14\pi \left( {l + 14} \right)\]
Dividing both sides of the equation by \[14\pi \], we get
\[\begin{array}{l} \Rightarrow \dfrac{{920}}{{14\pi }} = l + 14\\ \Rightarrow 20.92 = l + 14\end{array}\]
Subtracting 14 from both sides of the equation, we get
\[\begin{array}{l} \Rightarrow 20.92 - 14 = l + 14 - 14\\ \Rightarrow 6.92{\rm{ m}} = l\end{array}\]
Rounding to the nearest whole number, we get
\[\therefore l=7\text{ m}\]

Therefore, the slant height of the conical tent is 7 m.

Note:
We should remember the formulas for surface areas of a cone. We can make a mistake by using the incorrect formula, like using \[\pi rl\] for the total surface area of a cone. Also, we should always round off the answer to the nearest whole number at the end of the solution. In the above solution, we did not round off \[20.92\] to 21. Only the final value of slant height is rounded off. This helps in getting an accurate answer.