Question

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 Rs 500 per m^2. (Note that the base of the tent will not be covered with canvas.)

Answer 1

Given :

• The diameter of the cylindrical part = 4 m.
• The height of the cylindrical part = 2.1 m.
• The slant height of the top = 2.8 m.

To Find :

• The area of the canvas used for making a tent.
• The cost of the canvas of the tent at the rate of Rs. 500 per m².

Solution :

• Area of the canvas used for making a tent.

Area of the canvas = C.S.A. of cylindrical part + C.S.A. of conical top

We know that,

• C.S.A. of cylinder (Curved surface area) = 2πrh.
• C.S.A. of cone (Curved surface area) = πrl.

$\bf \implies 2\pi rh + \pi rl \\ \\ \\ \bf \implies \pi r(2h + l)$

Where,

• r = radius = d / 2 = 4 / 2 = 2 m.
• h = height = 2.1 m.
• l = slant height = 2.8 m.

$\bf \implies \dfrac{22}{7} \times 2 \: m \: (2 \times 2.1 \: m+ 2.8 \: m) \\ \\ \\ \bf \implies \dfrac{44}{7} \: m \times (4.2 \: m+ 2.8 \: m) \\ \\ \\ \bf \implies \dfrac{44}{7} \times 7.0 \: {m}^{2} \\ \\ \\\bf \implies \dfrac{44}{ \not7} \times \not7 \: {m}^{2} \\ \\ \\ \bf \implies 44 \: {m}^{2}$

Hence,

The total area of the canvas used for making a tent is 44 m².

• The cost of the canvas of the tent at the rate of Rs. 500 per m².

Cost of the canvas = A × R

Where,

• A = area = 44 m²
• R = rate = Rs. 500 per m²

$\bf \implies 44 \times Rs. \: 500 \\ \\ \\ \bf \implies Rs. \: 22000 \\ \\ \\ \: \: \bf \therefore \: \: Cost = Rs. \: 22000$

Hence,

The cost of the canvas is Rs. 22000.

Answer 2

It is known that a tent is a combination of cylinder and a cone.

From the question we know that

Diameter = 4 m

Slant height of the cone (l) = 2.8 m

Radius of the cone (r) = Radius of cylinder = 4/2 = 2 m

Height of the cylinder (h) = 2.1 m

So, the required surface area of tent = surface area of cone + surface area of cylinder

= πrl+2πrh

= πr(l+2h)

= (22/7)×2(2.8+2×2.1)

= (44/7)(2.8+4.2)

= (44/7)×7 = 44 m2

∴ The cost of the canvas of the tent at the rate of ₹500 per m2 will be

= Surface area × cost per m2

44×500 = ₹22000

So, Rs. 22000 will be the total cost of the canvas