Q7:

A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are $2.1$ m and $4$ m respectively, and the slant height of the top is $2.8$ m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of $₹ 500$ per m$^2$. (Note that the base of the tent will not be covered with canvas.)

Given:

• Shape of the tent: A cylinder surmounted by a cone.
• Height of the cylindrical part ($h_c$) = $2.1$ m.
• Diameter of the cylindrical part ($d$) = $4$ m.
• Slant height of the conical part ($l$) = $2.8$ m.
• Rate of canvas = $₹ 500$ per m$^2$.

To Find:

1. Total area of the canvas used for the tent.
2. Total cost of the canvas.

Step 1: Determine the radius of the base.

Since the diameter of the cylindrical part is $4$ m, the radius ($r$) is half of the diameter.

$r = \frac{d}{2} = \frac{4}{2} = 2$ m.

[Since the cone is mounted on the cylinder, the radius of the cone is also $r = 2$ m.]

Step 2: Calculate the surface area of the canvas.

The canvas covers the curved surface area of the cylinder and the curved surface area of the cone.

Formula for Curved Surface Area (CSA) of a cylinder = $2\pi rh_c$.

Formula for Curved Surface Area (CSA) of a cone = $\pi rl$.

Total Area ($A$) = $CSA_{cylinder} + CSA_{cone} = 2\pi rh_c + \pi rl = \pi r(2h_c + l)$.

Substituting the values ($r=2, h_c=2.1, l=2.8, \pi = \frac{22}{7}$):

$A = \frac{22}{7} \times 2 \times (2 \times 2.1 + 2.8)$

$A = \frac{44}{7} \times (4.2 + 2.8)$

$A = \frac{44}{7} \times 7$

$A = 44$ m$^2$.

Step 3: Calculate the total cost of the canvas.

Cost = Total Area $\times$ Rate per m$^2$.

Cost = $44 \times 500$.

Cost = $22,000$.

Final Answer: The total area of the canvas used is $44$ m$^2$ and the total cost of the canvas is $₹ 22,000$.