Question

Due to sudden floods, some welfare associations jointly requested the government to get 100 tents fixed immediately and offered to contribute 50% of the cost. If the lower part of each tent is of the form of a cylinder of diameter 4.2m and height 4m with the conical upper part of the same diameter but height 2.8m and the canvas to be used costs Rs.100 per sqm. Find the amount, the associations will have to pay. What values are shown by the associations? [Use, \[\pi =\dfrac{22}{7}\]]

Answer 1

Hint: Find the total surface area of the canvas by finding the curved surface area of cone and cylinder. Find the cost for 100 canvas tent, find how much the association would pay taking 50%

Complete step-by-step answer
We are given that the lower part of the tent is cylindrical in shape.

The diameter of the lower part of the tent = 4.2m.
Height of the lower part of the tent = 4m = H.
The upper part is conical in shape with the same diameter of the lower part.
The height of the conical upper part = 2.8m = h.
Check the figure, as the diameter = 4.2m.
Radius, r = diameter / 2\[=\dfrac{4.2}{2}=2.1\]m.
In the case of cones, we know that, \[{{l}^{2}}={{r}^{2}}+{{h}^{2}}\].
Where, l = slant height of cone.
\[\therefore l=\sqrt{{{r}^{2}}+{{h}^{2}}}\], we know r = 2.1 cm and h = 2.8m.
\[\begin{align}
  & l=\sqrt{{{\left( 2.1 \right)}^{2}}+{{\left( 2.8 \right)}^{2}}}=\sqrt{12.25} \\
 & l=3.5m \\
\end{align}\]
Thus we got the slant height of the cone as 2.5m.
The surface area of the canvas to be used = CSA of conical part + CSA of cylindrical part.
We know that the curved surface area (CSA) of a cone is \[\pi rl\].
Similarly, the CSA of a cylinder is \[2\pi rH\].
Thus, surface area of canvas = CSA of conical part + CSA of cylindrical part.
                                                   = \[\pi rl+2\pi rH\]
Let us substitute the values, r = 2.1m, l = 3.5m, H = 4m, \[\pi =\dfrac{22}{7}\].
SA of canvas \[=\dfrac{22}{7}\left[ 3.5\times 2.1+2\times 2.1\times 4 \right]=\dfrac{22}{7}\times 2.1\left[ 3.5+8 \right]\]
                       \[\begin{align}
  & =\dfrac{22}{7}\times 2.1\times 11.5 \\
 & =75.9{{m}^{2}} \\
\end{align}\]
\[\therefore \] Surface area of canvas = \[75.9{{m}^{2}}\].
We have been given the cost of canvas = Rs.100\[/{{m}^{2}}\].
\[\therefore \] Total cost of canvas to be used for 75.9\[{{m}^{2}}\] = \[75.9\times 100\] = Rs.75900.
The association is willing to pay 50% of the cost.
\[\therefore \] The amount that the association will have to pay for each tent
= 50% of 7590 \[=\dfrac{50}{100}\times 7590\] = Rs.3795.
Thus for each tent the association will pay Rs.3795.
So, the amount that the association will have to pay for 100 tents = Rs.3795 \[\times \] 100 = Rs.379500.
Hence, the amount that the associations will have to pay for 100 tents is Rs.3, 79, 500.
The associations are showing humanity and they are very helpful in nature.

Note: You should know the curved surface of the cylinder and cone to find the total surface area of the canvas. As we are constructing a tent, we need the area and not the outer perimeter or volume. To get the correct cost, you need to find an area, as the cost of canvas is given per area.