18
Case Study - 3
For the Kumbh Mela, Uttar Pradesh Government prescribed the following for the contractors to pitch the tents.
Each tent must be of cylindrical base of radius $21$ m and height $5$ m, surmounted by a conical part of height $20$ m. The cylindrical part must have a white coloured thick fabric costing ₹60 per square meter, while the conical part must have PVC coated blue fabric costing ₹70 per square meter.
Based on the above information, answer the following questions:
(a) How much blue PVC (in sq.m) is required and what will be its total cost?
(b) How much white fabric (in sq.m) is required and what will be its total cost?
For the Kumbh Mela, Uttar Pradesh Government prescribed the following for the contractors to pitch the tents.
Each tent must be of cylindrical base of radius $21$ m and height $5$ m, surmounted by a conical part of height $20$ m. The cylindrical part must have a white coloured thick fabric costing ₹60 per square meter, while the conical part must have PVC coated blue fabric costing ₹70 per square meter.
Based on the above information, answer the following questions:
(a) How much blue PVC (in sq.m) is required and what will be its total cost?
(b) How much white fabric (in sq.m) is required and what will be its total cost?
Show SolutionHide Solution↓
Given: Radius of cylindrical base $r = 21$ m.
Height of cylindrical part $h_c = 5$ m.
Height of conical part $h_k = 20$ m.
(a) For the conical part (blue PVC):
Slant height $l = \sqrt{r^2 + h_k^2} = \sqrt{21^2 + 20^2} = \sqrt{441 + 400} = \sqrt{841} = 29$ m.
Curved Surface Area (CSA) of conical part $= \pi r l = \frac{22}{7} \times 21 \times 29 = 22 \times 3 \times 29 = 66 \times 29 = 1914$ m$^2$.
Cost of blue PVC $= 1914 \times \text{Rs}70 = \text{Rs}133980$.
(b) For the cylindrical part (white fabric):
Curved Surface Area (CSA) of cylindrical part $= 2\pi r h_c = 2 \times \frac{22}{7} \times 21 \times 5 = 2 \times 22 \times 3 \times 5 = 660$ m$^2$.
Cost of white fabric $= 660 \times \text{Rs}60 = \text{Rs}39600$.
Height of cylindrical part $h_c = 5$ m.
Height of conical part $h_k = 20$ m.
(a) For the conical part (blue PVC):
Slant height $l = \sqrt{r^2 + h_k^2} = \sqrt{21^2 + 20^2} = \sqrt{441 + 400} = \sqrt{841} = 29$ m.
Curved Surface Area (CSA) of conical part $= \pi r l = \frac{22}{7} \times 21 \times 29 = 22 \times 3 \times 29 = 66 \times 29 = 1914$ m$^2$.
Cost of blue PVC $= 1914 \times \text{Rs}70 = \text{Rs}133980$.
(b) For the cylindrical part (white fabric):
Curved Surface Area (CSA) of cylindrical part $= 2\pi r h_c = 2 \times \frac{22}{7} \times 21 \times 5 = 2 \times 22 \times 3 \times 5 = 660$ m$^2$.
Cost of white fabric $= 660 \times \text{Rs}60 = \text{Rs}39600$.