A solid wooden toy is prepared by joining a cone, a cylinder and a sphere, as shown in the given diagram. The radius of each of the three solids is 7 cm and heights of each of the cone and the cylinder is 24 cm. Find :
(a) the total surface area of the given solid.
(b) the cost of painting the total surface at the rate of ₹ 0.50 per cm2.
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(a) Given,
Radius of each solid (r) = 7 cm
Height of cone = Height of cylinder = h = 24 cm
Let slant height of cone be l.
By formula,
$$\begin{aligned}l = \sqrt{r^2 + h^2} \\[1em] = \sqrt{7^2 + 24^2} \\[1em] = \sqrt{49 + 576} \\[1em] = \sqrt{625} \\[1em] = 25.\end{aligned}$$
Total surface area of sphere = 4πr2
= 4 × $\frac{22}{7}$ × 72
= 616 cm2.
Total surface area of cylinder = 2πr(h + r)
= 2 × $\frac{22}{7}$ × 7 × (24 + 7)
= 2 × 22 × 31
= 1364 cm2.
Total surface area of cone = πr(r + l)
= $\frac{22}{7}$ × 7 × (7 + 25)
= 22 × 32
= 704 cm2
Total surface area of toy = Total surface area of (sphere + cylinder + cone)
= 616 + 1364 + 704
= 2684 cm2.
Hence, total surface area of toy = 2684 cm2.
(b) Given,
Cost of painting = ₹ 0.50 per cm2.
The cost of painting the total surface = 2684 × ₹ 0.50 = ₹ 1,342
Hence, total cost of painting = ₹ 1,342.