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Question
The given figure shows the cross-section of a cone, a cylinder and a hemisphere all with the same diameter 10 cm and the other dimensions are as shown.
Calculate :
- the total surface area.
- the total volume of the solid and
- the density of the material if its total weight is 1.7 kg.
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Solution
Diameter = 10 cm
Therefore, radius (r) = 5 cm
Height of the cone (h) = 12 cm
Height of the cylinder = 12 cm
∴ `l = sqrt(h^2 + r^2)`
= `sqrt(12^2 + 5^2)`
= `sqrt(144 + 25)`
= `sqrt(169)`
= 13 cm
i. Total surface area of the solid
= `pirl + 2pirh + 2pir^2`
= `pir(l + 2h + 2r)`
= `22/7 xx 5[13 + (2xx12) + (2 xx 5)]`
= `110/7 [13 + 24 + 10]`
= `110/7 xx 47`
= `5170/7`
= 738.57 cm2
ii. Total volume of the solid
= `1/3pir^2h + pir^2h + 2/3pir^3`
= `pir^2 [1/3h + h + 2/3r]`
= `22/7 xx 5 xx 5[1/3 xx 12 + 12 + 2/3 xx 5]`
= `550/7 [4 + 12 + 10/3]`
= `550/7 [16 + 10/3]`
= `550/7 xx 58/3`
= `31900/21`
= 1519.0476 cm3
iii. Total weight of the solid = 1.7 kg
∴ Density = `"Mass"/"Volume"`
= `(1.7 xx 1000)/(1519.0476)` gm/cm3
= `(17 xx 1000 xx 10000)/(10 xx 15190476)` gm/cm3
= 1.119 gm/cm3
`=>` Density = 1.12 gm/cm3
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