Question

A hemispherical dome, open at the base is made from a sheet of fibre. If the radius of the hemispherical dome is $ 40\;cm $ and $ \dfrac{{13}}{{170}} $ of the fibre sheet was wasted in making the dome, then find the cost of the dome at the rate of $ 35 $ per $ 100\;c{m^2} $ .

Answer 1

Hint: Calculate the surface area of the hollow hemisphere and then use the rate of the fibre sheet to calculate the cost for the area of the hemisphere body to make. Use the relevant formula of hemisphere and not of sphere.

Complete step-by-step answer:
A hemispherical dome, open at the base is made from a sheet of fibre. If the radius of the hemispherical dome is $ 40\;cm $ and $ \dfrac{{13}}{{170}} $ of the fibre sheet was wasted in making the dome at the rate of $ 35 $ per $ 100\;c{m^2} $ .
The formula for the surface area of the hemisphere is equal to $ 2\pi {r^2} $ .
As per given the radius of the hemisphere is equal to $ 40\;cm $ . Now calculate the volume of the hemisphere.
 $
\Rightarrow A = 2\pi {r^2} \\
   = 2 \times 3.14 \times {\left( {40\;cm} \right)^2} \\
   = 10048\;c{m^2} \;
  $
The surface area of the hemisphere is equal to $ 10048\;c{m^2} $ .
Assume that the area of fibre sheet used is equal to $ x $ . As given that $ \dfrac{{13}}{{170}} $ of the fibre sheet was wasted in making the dome. So,
 $
\Rightarrow 10048 + \dfrac{{13}}{{170}}x = x \\
\Rightarrow 10048 = x - \dfrac{{13}}{{170}}x \\
\Rightarrow 10048 = \dfrac{{157}}{{170}}x \\
\Rightarrow x = 10048 \times \dfrac{{170}}{{157}} \\
\Rightarrow x = 10880 \;
  $
As the total area of fibre sheet used in making the dome is equal to $ 10880\;c{m^2} $ .
The cost of a fibre sheet for each $ 100\;c{m^2} $ is equal to $ 35 $ rupees. The cost of fibre sheet for each $ 1\;c{m^2} $ is equal to $ \dfrac{{35}}{{100}} $ rupees and the cost for $ 10880\;c{m^2} $ is equal to $ \dfrac{{35}}{{100}} \times 10880 = 3808 $ rupees.
So, the correct answer is “3808 Rs”.

Note: Out of the total sheet used in making the dome $ \dfrac{{13}}{{170}} $ of the fibre sheet is wasted in making the dome and calculate the area of sheet that is used to make the dome only not the wastage part and find the cost for it.