MP Board · Class 10 · Mathematics · Surface Areas and VolumesA 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.)

Step-by-Step Solution:

1. Given Data:

• Diameter of the cylindrical part ($d$) = 4 m
• Radius of the cylindrical part ($r$) = $\frac{4}{2} = 2$ m
• Height of the cylindrical part ($h$) = 2.1 m
• Slant height of the conical top ($l$) = 2.8 m

2. Radius of the Conical Top:

• Since the cone is surmounted on the cylinder, the radius of the cone is the same as the radius of the cylinder, i.e., $r = 2$ m.

3. Formulae Used:

• Curved Surface Area (CSA) of cylinder = $2\pi rh$
• Curved Surface Area (CSA) of cone = $\pi rl$
• Total area of canvas = CSA of cylinder + CSA of cone

4. Calculation:

• CSA of cylinder = $2 \times \frac{22}{7} \times 2 \times 2.1$ $= 2 \times \frac{22}{7} \times 4.2$ $= 2 \times 22 \times 0.6 = 26.4$ $m^2$

• CSA of cone = $\frac{22}{7} \times 2 \times 2.8$ $= \frac{22}{7} \times 5.6$ $= 22 \times 0.8 = 17.6$ $m^2$

• Total area of canvas = $26.4 + 17.6 = 44$ $m^2$

5. Cost Calculation:

• Rate of canvas = Rs 500 per $m^2$
• Total Cost = Total Area $\times$ Rate $= 44 \times 500$ = Rs 22,000

Answer:\nThe area of the canvas used is $44$ $m^2$ and the total cost is Rs 22,000.