Surface Areas and Volumes Test

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Surface Areas and Volumes Test - 31

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Weekly Quiz Competition

Question 1
1 / -0

The height of the frustum cone is 30 cm. The area of the top and bottom circle of the cone is 6 cm$$^2$$ and 96 cm$$^2$$ respectively. Find its volume.

A
1,260 cm$$^3$$

B
1,700 m$$^3$$

C
1,260 m$$^3$$

D
1,260 cm

Solution

Volume of the frustum cone is V = $$\displaystyle{\frac{h}{3}}[\pi R^2 + \pi r^2 + \sqrt{(\pi R^2)(\pi r^2)}$$]
Area of the top cone = 6 cm$$^2$$
Area of the bottom cone = 96 cm$$^2$$
Height = 30 cm
Volume = $$\displaystyle{\frac{30}{3}}[96 + 6 + \sqrt{96x6}]$$
$$= 10[102 + \sqrt{576}]$$
$$= 10[102 + 24]$$
$$= 10 \times 126$$
$$=$$ 1,260 cm$$^3$$
Question 2
1 / -0

The height of the frustum cone is 210 m. The area of the top and bottom circle of the cone is 10 m$$^2$$ and 250 m$$^2$$ respectively. Find its volume.

A
20,700 m$$^3$$

B
21,700 m$$^3$$

C
22,700 m$$^3$$

D
23,700 m$$^3$$

Solution

Volume of the frustum cone is V = $$\displaystyle{\frac{h}{3}}[\pi R^2 + \pi r^2 + \sqrt{(\pi R^2)(\pi r^2)}$$]
Area of the top cone = 10 m$$^2$$
Area of the bottom cone = 250 m$$^2$$
Height = 210 m
Volume = $$\displaystyle{\frac{210}{3}}$$
= 70[260 + $$\sqrt{2500}$$]
= 70[260 + 50]
= 70 x 310
= 21,700 m$$^3$$
Question 3
1 / -0

The volume of the frustum of a cone is 54 cm$$^3$$ and its height is 6 cm, bottom radius, R = 1 cm. Find its top radius, r. (Use $$\pi$$ = 3).

A
2 cm

B
3 cm

C
4 cm

D
5 cm

Solution

Volume of the frustum cone is $$V = \displaystyle{\frac{\pi h}{3}}(R^2 + Rr + r^2)$$
Radius, R = 1 cm, r = ?, h = 6 cm
Volume = 54 cm$$^3$$
$$54 = (3 \times 6 )/(3)[1^2 + 1 \times r + r^2]$$
$$54 = 6[1 + r + r^2]$$
$$9 = 1 + r + r^2$$
$$r^2 + r - 8 = 0$$
On factoring neglect the negative value of r, we get
r = 2 cm
Question 4
1 / -0

The curved surface area of frustum cone is 400 $$m^2$$ The diameter of a cone is 0.5 and 2 m. Find the slant height. (Use $$\pi$$ = 3.14).

A
123.47 m

B
124.47 m

C
101.91 m

D
126.47 m

Solution

Curved surface area = $$\pi$$(R + r)s
Diameter = radius $$\times 2$$
Radius, R = 0.25 m, r = 1 m
400 = $$3.14 \times (0.25 + 1) \times s$$
400 = $$3.14 \times (1.25) \times s$$
400 = 3.925 $$\times s$$
S = 400/3.925
Slant Height = 101.91 m
Question 5
1 / -0

Find the curved surface area of frustum cone radii 0.2 and 3.2 cm and a slant height 5 cm. (Use $$\pi$$ = 3.14).

A
53.38 $$cm^2$$

B
52.38 $$cm^2$$

C
51.38 $$cm^2$$

D
50.38 $$cm^2$$

Solution

Curved surface area = $$\pi(R + r)s$$
= $$\pi (3.2 + 0.2) \times 5$$
= $$3.14 \times (3.4) \times 5$$
= $$3.14 \times 17$$
= 53.38 $$ cm^2$$
Question 6
1 / -0

Find the curved surface area of frustum cone radii 0.5 and 5.2 m and a slant height 10 m. (Use $$\pi$$ = 3.14).

A
176.98 $$m^2$$

B
178.98 $$m^2$$

C
177.98 $$m^2$$

D
179.98 $$m^2$$

Solution

Curved surface area = $$\pi$$(R + r)s
= $$\pi (0.5 + 5.2) \times 10$$
= $$\pi (5.7) \times 10$$
= $$3.14 \times (5.7)$$
= 178.98 $$m^2$$
Question 7
1 / -0

The volume of the frustum of a cone is $$600 m$$$$^3$$ and its height is $$12 m$$, bottom radius, $$R = 2 m$$. Find its top radius, $$r$$. (Use $$\pi$$ = 3.14).

A
$$5.7$$ m

B
$$4.9$$ m

C
$$3.5$$ m

D
$$6.2$$ m

Solution

Volume of frustum $$=\dfrac { 1 }{ 3 } \pi \left( { R }^{ 2 }+{ r }^{ 2 }+Rr \right) \times h$$
$$\Rightarrow \quad 600{ m }^{ 3 }=\dfrac { 1 }{ 3 } \times 3.14\left( { 2 }^{ 2 }+{ r }^{ 2 }+2r \right) \times 12$$
$$\Rightarrow \quad { r }^{ 2 }+2r+4=48\Rightarrow { r }^{ 2 }+2r-44=0\Rightarrow r=\left( -1\pm 3\sqrt { 5 } \right) $$
$$\Rightarrow \quad r=\quad -1-3\sqrt { 5 } ,\quad -1+3\sqrt { 5 } $$
Neglecting negative value, $$\therefore \quad r=-1+3\sqrt { 5 } =6.70-1=5.7m$$
Question 8
1 / -0

The volume of frustum of a cone is 600 m$$^3$$. The area of the bottom circle of the cone is 10 m$$^2$$ and the area of the top circle of the cone is 10 m$$^2$$. Find its height.

A
60 m

B
70 m

C
50 m

D
40 m

Solution

Volume of the frustum cone is V = $$\displaystyle{\frac{h}{3}}[A_1 + A_2 + \sqrt{A_1 A_2}$$]
Area of the bottom cone, A$$_1$$ = 10 m$$^2$$
Area of the top cone, A$$_2$$ = 10 m$$^2$$
Height = ?
Volume = 600 m$$^3$$
$$600 = h/3[10 + 10 + \sqrt{10 \times 10}]$$
$$600 \times 3 = h[20 + \sqrt{100}]$$
$$1,800 = h[20 + 10]$$
$$1,800/30 = h$$
$$h = 60 m$$
Question 9
1 / -0

The volume of frustum of a cone is 3,600 cm$$^3$$. The area of the bottom circle of the cone is 4 cm$$^2$$ and the area of the top circle of the cone is 16 cm$$^2$$. Find its height.

A
385.7 cm

B
375.7 cm

C
365.7 cm

D
355.7 cm

Solution

Volume of the frustum cone is V = $$\displaystyle{\frac{h}{3}}[A_1 + A_2 + \sqrt{A_1 A_2}$$]
Area of the bottom cone, $$A_1 = 4 \,cm^2$$
Area of the top cone, $$A_2 = 16\, cm^2$$
Height = ?
Volume = 3,600 cm
$$3,600 = h/3[4 + 16 + \sqrt{4 \times 16}]$$
$$3,600 \times 3 = h[20 + \sqrt{64}]$$
$$10,800 = h[20 + 8]$$
$$10,800/28 = h$$
$$h = 385.7 cm$$
Question 10
1 / -0

The curved surface area of frustum cone is $$303.3$$ $$in^2$$. The diameter of a top cone is $$0.5$$ and the radius of the bottom cone is $$0.2$$ in. Find the slant height. (Use $$\pi$$ = 3.14).

A
281.49 in

B
282.49 in

C
214.64 in

D
285.49 in

Solution

Curved surface area = $$\pi(R + r)s$$
Diameter $$=$$ radius $$\times 2$$
Radius, R $$= 0.25$$ in, r $$= 0.2$$ in
$$\Rightarrow 303.3 =$$ $$3.14 \times (0.25 + 0.2) \times s$$
$$\Rightarrow 303.3 =$$ $$3.14 \times (0.45) \times s$$
$$\Rightarrow \dfrac {303.3}{3.14} \times 0.45$$ $$= s$$
Slant Height $$=s= 214.64$$ in

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Surface Areas and Volumes Test - 31

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Surface Areas and Volumes Test - 31

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SHARING IS CARING

Question 1

1,260 cm$$^3$$

1,700 m$$^3$$

1,260 m$$^3$$

1,260 cm

Solution

Question 2

20,700 m$$^3$$

21,700 m$$^3$$

22,700 m$$^3$$

23,700 m$$^3$$

Solution

Question 3

2 cm

3 cm

4 cm

5 cm

Solution

Question 4

123.47 m

124.47 m

101.91 m

126.47 m

Solution

Question 5

53.38 $$cm^2$$

52.38 $$cm^2$$

51.38 $$cm^2$$

50.38 $$cm^2$$

Solution

Question 6

176.98 $$m^2$$

178.98 $$m^2$$

177.98 $$m^2$$

179.98 $$m^2$$

Solution

Question 7

$$5.7$$ m

$$4.9$$ m

$$3.5$$ m

$$6.2$$ m

Solution

Question 8

60 m

70 m

50 m

40 m

Solution

Question 9

385.7 cm

375.7 cm

365.7 cm

355.7 cm

Solution

Question 10

281.49 in

282.49 in

214.64 in

285.49 in

Solution

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