Question

The circumference of the base of a 24 m high conical tent is 44 m. Calculate the length of a canvas used in making the tent if the width of the canvas is 2 m.
A. 257 m
B. 275 m
C. 752 m
D. 285 m

Answer 1

Hint: Here, we will use the formula of circumference \[2\pi r\] to find the radius of the circle. Then we will find the slant height \[l = \sqrt {{h^2} + {r^2}} \] to find the surface area of the canonical tent \[\pi rl\]. Then the surface area is divided by the width for the length of required canvas.

Complete step-by-step solution
Given that the height \[h\] is 24 m.
Let the radius of the base of a conical tent is \[r\] meters and the slant height be \[l\] meters.

We know that the circumference of a circle is \[2\pi r\], where \[r\] is the radius of the circle.

Since we are given that the circumference of the conical tent is 44 m.

\[ \Rightarrow 2\pi r = 44\]

Dividing the above equation by \[2\pi \] on each of the sides, we get

\[
   \Rightarrow \dfrac{{2\pi r}}{{2\pi }} = \dfrac{{44}}{{2\pi }} \\
   \Rightarrow r = \dfrac{{44}}{{2\pi }} \\
\]

Substituting the value of \[\pi \] in the above equation and simplify it, we get

\[
   \Rightarrow r = \dfrac{{44}}{{2 \times \dfrac{{22}}{7}}} \\
   \Rightarrow r = \dfrac{{44 \times 7}}{{2 \times 22}} \\
   \Rightarrow r = 7{\text{ m}} \\
\]

We will now find the slant height \[l\] by substituting the values of \[h\] and \[r\] in the equation \[l = \sqrt {{h^2} + {r^2}} \].

\[
   \Rightarrow l = \sqrt {{7^2} + {{24}^2}} \\
   \Rightarrow l = \sqrt {49 + 576} \\
   \Rightarrow l = \sqrt {625} \\
   \Rightarrow l = 25{\text{ m}} \\
\]

Substituting the above values of \[r\] and \[l\] in the surface area \[\pi rl\], we get

\[
  {\text{Surface area}} = \pi rl \\
   = \dfrac{{22}}{7} \times 7 \times 25 \\
   = 22 \times 25 \\
   = 550{\text{ }}{{\text{m}}^2} \\
\]

We know that the area of canvas is equal to the surface area of the cone.
Thus, the area of the canvas is 550 m\[^2\].

Using the width of the canvas is 2 m, we will now find the length of the canvas by dividing the area of canvas by 2.

\[
  {\text{Length of canvas}} = \dfrac{{{\text{Area of canvas}}}}{2} \\
   = \dfrac{{550}}{2} \\
   = 275{\text{ m}} \\
\]

Thus, the length of canvas used in making the tent is 275 m.

Hence, the option B is correct.

Note: In this question, students should be familiar with formulae of circumference of a circle, \[2\pi r\] and the surface area of cone, \[\pi rl\]. We will use the equality of area of canvas and the surface area of the cone for the correct solution. We have to divide the area of the canvas by the width of the canvas to find the length of the canvas.