Question

A circle tent is cylindrical up to a height of ${\text{3m}}$ and conical above it. If the diameter of the base is ${\text{12m}}$ and the slant height of the conical part is ${\text{10m}}$, then the total area of the canvas just exceeds
A.$200{m^2}$
B.$250{m^2}$
C.$300{m^2}$
D.$350{m^2}$

Answer 1

Hint: We can divide the tent into two parts. Then we can find a curved surface area of the cylindrical part using the height and diameter and the curved surface area of the cone using the slant height and the diameter. Then we add these two areas to get the total surface area and this area is the area of the canvas.

Complete step by step answer:

To find the area of the canvas of the tent, we find the total surface area of the tent. For that, we can divide the tent into a cylinder and a cone and find their area separately.
Consider the cylindrical part of the tent. Its height is given as ${\text{h = 3m}}$. It has diameter${\text{12m}}$. So, the radius is given by,
$r = \dfrac{{diameter}}{2} = \dfrac{{12}}{2} = 6m$
Then curved surface area of the cylinder is given by,
${A_1} = 2\pi rh$
Applying the values of height and radius, we get,
${A_1} = 2 \times 3.14 \times 6 \times 3$
After multiplication, we get,
${A_1} = 113.04{m^2}$
So, the curved surface area of the cylinder is $113.04{m^2}$
Now we consider the conical part. It has slant height of ${\text{l = 10m}}$ and the radius is same as that of the cylinder, i.e. $r = 6m$.
Then the curved surface area of the cone is given by,
${A_2} = \pi rl$
Applying the values of height and radius, we get,
${A_2} = 3.14 \times 6 \times 10$
After multiplication, we get,
${A_2} = 188.4{m^2}$
So, the curved surface area of the conical part is $188.4{m^2}$.
So, the total surface area is given by sum of the area of the cylinder and cone. So, we get,
$A = {A_1} + {A_2}$
On applying the values and adding, we get,
$A = 113.04 + 188.4 = 301.44{m^2}$
Thus, the surface area of the tent is $301.44{m^2}$. So, the area of the canvas is also the same.
Therefore, the area of the canvas just exceeds ${\text{300}}{{\text{m}}^{\text{2}}}$.
So, the correct answer is option C.

Note: The area of the canvas used will be equal to the surface area of the tent. The property of surface areas of 3-dimensional objects is used in the problem. The surface area of an object is the sum of the area of its surfaces. We are considering only the curved surface area of the cylinder as the base of the cylinder is not covered with canvas and the top is covered by a cone. Similarly, the cone has the cylinder at the base, so we are taking only its curved surface area.