Question

A hemisphere dome of a building needs to be painted. If the circumference of the base of the dome is 44 m, find the cost of painting it, given the cost of painting is Rs 2 per 100 \[{\text{c}}{{\text{m}}^2}\] .
A. Rs 62600
B. Rs 63000
C. Rs 61000
D. Rs 61600

Answer 1

Hint: To paint the dome, we have to find the surface area of the dome. Since the circumference of the base is given so by putting values in the formula we will get the radius of the dome.

Complete step-by-step answer:
Since only the rounded surface of the dome is to be painted so we need to find the curved surface area of the hemisphere to know the extent of painting to be done.

Now, circumference of the base of the dome is 44m
We know that circumference or perimeter of circle = $2\pi r$, r is the radius of the circle.
Therefore, $2\pi r = 44$m [ r = radius of the dome]
Or $r = \dfrac{{44 \times 7}}{{2 \times 22}} = 7$
Therefore, the radius of the dome is 7 m.
Now, the curved surface area of the dome = $2\pi {r^2}$
Curved surface area of the dome = $2 \times \dfrac{{22}}{7} \times 7 \times 7 = 308{{\text{m}}^2}$
Now, calculation for cost of painting the surface area of the dome:
Now, cost of painting 100 \[{\text{c}}{{\text{m}}^2}\] is Rs 2
So, cost of painting 1 \[{{\text{m}}^2}\] is Rs 200
[Since to paint 10 cm × 10 cm we need Rs 2, so to paint 100 cm × 100 cm we need Rs 200 i.e. to paint 1 m × 1 m = 1 \[{{\text{m}}^2}\] we need Rs 200]
Therefore, Cost of painting the whole dome = 200 × 308 = Rs 61600

Hence, option (D) is correct.

Note: Students should keep in mind that Curved surface area of a Hemisphere is the area of the outer surface of the hemisphere called Curved Surface Area. Curved Surface area of Hemisphere is $2\pi {r^2}$.While Total Surface Area of Hemisphere is the area of the curved surface area and the area of the circle (base) is called Total Surface area. Total Surface area = $3\pi {r^2}$.To solve this type of question we must have a clear idea about what is asked in question and which surface area is required, also we should know the formula of hemisphere. In this question the curved surface area is to be painted.