Question

A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is 104 cm and the radius of each of the hemispherical ends is 7 cm, find the cost of polishing its surface at the rate of Rs 10 per dm².

Answer 1

Given: Total length of the solid \[L = 104\ \mathrm{cm}\]

Radius of hemispherical ends and cylinder \[r = 7\ \mathrm{cm}\]

Rate of polishing = \[₹,10\ \mathrm{per\ dm^{2}}\]

To find: Cost of polishing the surface

Formula: Height of cylindrical part: \[h = L - 2r\]

Total surface area of the solid:

\[S = \text{CSA of cylinder} + 2 \times (\text{CSA of hemisphere})\] 

\[S = 2\pi rh + 2(2\pi r^{2}) = 2\pi r(h + 2r) = 2\pi r L\] 

\[\text{Cost} = S\ (\text{in } \mathrm{dm^{2}}) \times \text{Rate}\]

Substitution: \[h = 104 - 2(7) = 90\ \mathrm{cm}\]

\[S = 2 \times \dfrac{22}{7} \times 7 \times (90 + 14) = 44 \times 104\]

Calculation: \[S = 4576\ \mathrm{cm^{2}}\] 

Since \[1\ \mathrm{dm} = 10\ \mathrm{cm}\],

\[1\ \mathrm{dm^{2}} = 100\ \mathrm{cm^{2}}\]: 

\[S = \dfrac{4576}{100}\ \mathrm{dm^{2}} = 45.76\ \mathrm{dm^{2}}\]

\[\text{Cost} = 45.76 \times 10 = ₹,457.60\]

Answer: \[\text{Cost of polishing} = ₹,457.60\]