Area and Volume of Combined Solid Objects

Question 1

A figure of a pencil is given aside. Find the total surface area and volume of the pencil.

Solution:

Here, radius of the base (r) = 7 cm

Height of the cylinder (h₁) = 39 cm

Height of the cone (h₂) = 24 cm

Total surface area = ?

Volume = ?

We know, $l^2 = h^2 + r^2 = (24)^2 + (7)^2 = 576 + 49 = 625$ cm²

Therefore, the slant height of the cone (l) = 25 cm

Again, total surface area

$= \pi r^2 + 2\pi r h_1 + \pi r l$

$= \pi r (r + 2h_1 + l)$

$= \frac{22}{7} \times 7 , (7 + 2 \times 39 + 25) = 22 \times 110 = 2420$ cm²

Volume (V) = volume of cylinder + volume of cone

$= \pi r^2 h_1 + \frac{1}{3}\pi r^2 h_2$

$= \pi r^2 \left(h_1 + \frac{1}{3} h_2\right) = \frac{22}{7} \times (7)^2 \left(39 + \frac{1}{3} \times 24\right) = 7238$ cm³

Question 2

A figure of ice cream is given aside. If the radius of the circular base is 21 cm and the volume of ice cream is 32340 cm³,

1. Find the height of conical part.
2. Find the total surface area.

Solution:

Here, radius of the base (r) = 21 cm

Volume of the cone with ice cream (V) = 32340 cm³

(a) We know,

Volume of cone with ice cream (V) = $\frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3$

$\text{or, } 32340 = \frac{1}{3}\pi r^2 (h + 2r)$

$\text{or, } 32340 = \frac{1}{3} \times \frac{22}{7} \times (21)^2 (h + 2 \times 21)$

$\text{or, } \frac{32340 \times 21}{22 \times 441} = (h + 42)$

$\text{or, } 70 - 42 = h$

∴ h = 28 cm

(b) Slant height (l)

$= \sqrt{h^2 + r^2}$

$= \sqrt{(28)^2 + (21)^2}$

$= \sqrt{1225}$

$= 35$ cm

Total surface area (TSA)

$= \pi r l + 2\pi r^2$

$= \pi r (l + 2r)$

$= \frac{22}{7} \times 21 , (35 + 2 \times 21)$

$= 66 , (35 + 42) = 66 \times 77$

$= 5082$ cm²

Question 3

The solid object in the given figure contains two cones. On the basis of the given measurements, find its volume.

Solution:

Here, the common diameter to the base to the both cones (d) = 6 cm

Total height of the solid = 20 cm

Let the height of cone of the left part = h₁ cm and the height of cone of the right part = h₂ cm

∴ $h_1 + h_2 = 20$ cm

(a) Radius of base (r) = $\frac{d}{2} = \frac{6}{2} = 3$ cm

Volume of solid (V)

$= \frac{1}{3}\pi r^2 (h_1 + h_2)$

$= \frac{1}{3} \times \frac{22}{7} \times (3)^2 (20) = 188.57$ cm³

Question 4

A figure of stupa is given aside. The shape of its lower part is square based prism and the upper part is square based pyramid. Then, find its

1. Volume
2. Total surface area.

Solution:

Here, total height of the stupa = 5.5 m

Height of the prism (h₁) = 5 m

Height of the pyramid (h₂) = 5.5 − 5 = 0.5 m

Length of the base of stupa (a) = 2 m

(a) Area of base (A₁) = $a^2 = 2^2 = 4$ m²

Volume of the prism shaped part (V₁) = $A_1 \times h_1 = 4 \times 5 = 20$ m³

Volume of the pyramid shaped part (V₂) = $\frac{1}{3} A_1 \times h_2 = \frac{1}{3} \times 4 \times 0.5 = \frac{2}{3}$ m³

Volume of the stupa (V) = $V_1 + V_2 = 20 + \frac{2}{3} = 20.67$ m³

(b) Perimeter of base (P) = $4a = 4 \times 2 = 8$ m

Slant height of the pyramidal part (l)

$= \sqrt{(h_2)^2 + \left(\frac{a}{2}\right)^2}$

$= \sqrt{(0.5)^2 + \left(\frac{2}{2}\right)^2} = \sqrt{0.25 + 1} = \sqrt{1.25}$ m

Lateral surface area of the prism shaped part (A₂) = $p \times h_1 = 8 \times 5 = 40$ m²

Lateral surface area of the pyramid shaped part (A₃) = $2al = 2 \times 2 \times \sqrt{1.25} = 4.47$ m²

Total surface area of the stupa (A)

$= A_1 + A_2 + A_3$

$= 4 + 40 + 4.47$

$= 48.47$ m²

Hence, the total surface area of the stupa is 48.47 m²

Question 5

The given figure contains two square based pyramids with equal height. If the length of the base of each pyramid is 6 cm and the total volume is 96 cm³, find the height of each pyramid.

Solution:

Here, the length of the base of each pyramid (a) = 6 cm

Total volume of pyramids (V) = 96 cm³

Suppose, the height of each pyramid = h

Total volume of pyramids (V) = $\frac{1}{3}a^2 h + \frac{1}{3} a^2 h$

$\text{or, } 96 = \frac{2}{3} a^2 h$

$\text{or, } \frac{96 \times 3}{2} = (6)^2 \times h$

$\text{or, } \frac{96 \times 3}{2 \times 36} = h$

∴ h = 4 cm

Hence, the height of each pyramid (h) is 4 cm.