Surface Areas and Volumes Test

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Surface Areas and Volumes Test - 39

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Weekly Quiz Competition

Question 1
1 / -0

Calculate the curved surface area of a frustum cone given below :

A
2,067.6 $$\text{m}^2$$

B
1,067.6 $$\text{m}^2$$

C
1,167.6 $$\text{m}^2$$

D
1,057.6 $$\text{m}^2$$

Solution

Given, $$r=5$$ m, $$R=15$$ m, $$l=17$$ m
We know curved surface area $$=$$ $$ \pi (R + r) l $$
$$=$$ $$\pi (15 + 5) \times 17$$
$$=$$ $$3.14 \times 20 \times 17$$
$$=1067.6$$ sq. m
Question 2
1 / -0

Calculate the curved surface area of a frustum cone given below :

A
1,120.5 $${cm}^2$$

B
1,220.5 $${cm}^2$$

C
1,320.5 $${cm}^2$$

D
1,020.5 $${cm}^2$$

Solution

Curved surface area = $$ \pi (R + r) \sqrt{(R-r)^2 + h^2} $$
= $$\pi (15 + 10) \sqrt{ (15 - 10)^2 + 12^2} $$
= $$3.14 \times 25 \times \sqrt{25 + 144}$$
= $$3.14 \times 25 \times 13$$
= 1,020.5 $${cm}^2$$
Question 3
1 / -0

A vessel is in the form of a frustum of a cone. Its radius at top end is 8 cm and the bottom end is 12 cm. Its height is 21 cm. Find the volume of the frustum cone.

A
1,128 $$\pi {cm}^3$$

B
3,128 $$\pi {cm}^3$$

C
4,128 $$\pi {cm}^3$$

D
2,128 $$\pi {cm}^3$$

Solution

Volume = $$\displaystyle \frac{\pi h}{3} (R^2 + Rr + r^2)$$
= $$\displaystyle \frac{\pi \times 21}{3} ({12}^2 + 12 \times 8 + 8^2)$$
= $$7 \pi (144 + 96 + 64)$$
= $$7 \pi (304)$$
= $$2,128 cm^3$$
Question 4
1 / -0

Calculate the total surface area of a frustum cone:
Given D = 1 cm, d = 0.1 cm, h = 5 cm.

A
18.298 $${cm}^2$$

B
17.298 $${cm}^2$$

C
16.298 $${cm}^2$$

D
11.159 $${cm}^2$$

Solution

Total surface area = $$\pi (R+ r) \sqrt{(R-r)^2 + h^2} + \pi R^2 + \pi r^2$$
Diameter = radius $$\times 2$$
R = 0.5 cm, r = 0.05 cm
= $$3.14 (0.5 + 0.05) \sqrt{(0.5 - 0.05)^2 + 5^2} + \pi {0.5}^2 + \pi {0.05}^2$$
= $$3.14 \times (0.550) \times 5.020 + 0.25 \times 3.14 + 0.0025 \times 3.14$$
= $$3.14 [2.761 + 0.785 + 0.00785]$$
= $$11.159 {cm}^2$$
Question 5
1 / -0

Calculate the curved surface area of a frustum cone:
Given D = 12 cm, d = 5 cm, s = 23 cm.

A
613.87 $${cm}^2$$

B
555.48 $${cm}^2$$

C
455.48 $${cm}^2$$

D
655.48 $${cm}^2$$

Solution

Curved surface area = $$\pi (R+ r)s$$
Diameter = radius $$\times 2$$
R = 6 cm, r = 2.5 cm
= $$\pi (6 + 2.5) \times 23$$
= $$3.14 \times 8.5 \times 23$$
= $$613.87 {cm}^2$$
Question 6
1 / -0

Calculate the total surface area of frustum cone given below:

A
4.05 $$\pi m^2$$

B
5.05 $$\pi m^2$$

C
6.05 $$\pi m^2$$

D
7.05 $$\pi m^2$$

Solution

Total surface area = $$\pi (R+r)s + \pi R^2 + \pi r^2$$
$$= \pi (0.2+ 0.1)20 + \pi 0.2^2 + \pi 0.1^2$$
$$= 6 \pi + 0.04\pi + 0.01 \pi $$
= $$6.05 \pi m^2$$
Question 7
1 / -0

Calculate the curved surface area of a frustum cone given below :

A
188.23 $$m^2$$

B
168.23 $$m^2$$

C
178.23 $$m^2$$

D
198.23 $$m^2$$

Solution

Curved surface area = $$ \pi (R + r) \sqrt{(R-r)^2 + h^2} $$
= $$\pi (5 + 0.1) \sqrt{ (5 - 0.1)^2 + 10^2} $$
= $$3.14 \times 5.1 \times \sqrt{24.01 + 100}$$
= $$3.14 \times 5.1 \times 11.13$$
= 178.23 $${m}^2$$
Question 8
1 / -0

A vessel is in the form of a frustum of a cone. Its radius at top end is 12 m and the bottom end is 10 m. Its volume is 369 $$ \pi m^3$$. Find its height.

A
3.64 m

B
3.04 m

C
3.54 m

D
3.44 m

Solution

Volume of a frustum cone = $$ \displaystyle \frac{\pi h}{3} (R^2 + Rr + r^2)$$
$$369 \pi$$$$= \displaystyle \frac{\pi \times h}{3} ({12}^2 + 12 \times 10 + {10}^2)$$
$$369 \times 3$$ = $$h(144 + 120 + 100)$$
$$1,107 = h (364)$$
h = 3.04 m
Question 9
1 / -0

The curved surface area of a frustum cone is 240 cm$$^2$$. The larger circle area is 50 cm$$^2$$. The total surface area is 750 cm$$^2$$. Find the smaller circle area of a cone.

A
560 cm$$^2$$

B
460 cm$$^2$$

C
660 cm$$^2$$

D
760 cm$$^2$$

Solution

Curved surface area of a frustum cone = $$\pi$$(r + R)s $$\Rightarrow$$ 240 cm$$^2$$
Total surface area of a frustum of cone = $$\pi$$(r + R)s +$$\pi$$r$$^2$$ + $$\pi$$R$$^2$$ $$\Rightarrow$$ 750 cm$$^2$$
Larger circle area = $$\pi$$R$$^2$$ $$\Rightarrow$$ 50 cm$$^2$$
Smaller circle area = $$\pi$$r$$^2$$ $$\Rightarrow$$ ?
Therefore,

SA = $$\pi$$(r + R)s +$$\pi$$r$$^2$$ + $$\pi$$R$$^2$$
$$750 = 240 + $$Area of smaller circle + 50
$$750 - 240 - 50$$ = Area of smaller circle
Area of smaller circle of a frustum cone = 460 cm$$^2$$
Question 10
1 / -0

The total surface area of a frustum cone is 311.56 m$$^2$$. The area of larger and smaller circle is 12 and 8.4 m$$^2$$. Find the curved surface area of a cone.

A
191.16 m$$^2$$

B
391.16 m$$^2$$

C
491.16 m$$^2$$

D
291.16 m$$^2$$

Solution

Curved surface area of a frustum cone = $$\pi$$(r + R)s $$\Rightarrow$$  ?
Total surface area of a frustum of cone = $$\pi$$(r + R)s+ $$\pi$$r$$^2$$ + $$\pi$$R$$^2$$ $$\Rightarrow$$  311.56 m$$^2$$
Smaller circle area = $$\pi$$r$$^2$$ $$\Rightarrow$$  8.4 m$$^2$$
Larger circle area = $$\pi$$R$$^2$$ => 12 m$$^2$$
Therefore,

SA = $$\pi$$(r + R)s +$$\pi$$r$$^2$$ + $$\pi$$R$$^2$$
311.56 = Curved surface area + 8.4 + 12
$$311.56 - 8.4 - 12 = $$Curved surface area
Curved surface area of a frustum cone = 291.16 m$$^2$$

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Re-Attempt Weekly Quiz Competition

Surface Areas and Volumes Test - 39

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Surface Areas and Volumes Test - 39

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Question 1

2,067.6 $$\text{m}^2$$

1,067.6 $$\text{m}^2$$

1,167.6 $$\text{m}^2$$

1,057.6 $$\text{m}^2$$

Solution

Question 2

1,120.5 $${cm}^2$$

1,220.5 $${cm}^2$$

1,320.5 $${cm}^2$$

1,020.5 $${cm}^2$$

Solution

Question 3

1,128 $$\pi {cm}^3$$

3,128 $$\pi {cm}^3$$

4,128 $$\pi {cm}^3$$

2,128 $$\pi {cm}^3$$

Solution

Question 4

18.298 $${cm}^2$$

17.298 $${cm}^2$$

16.298 $${cm}^2$$

11.159 $${cm}^2$$

Solution

Question 5

613.87 $${cm}^2$$

555.48 $${cm}^2$$

455.48 $${cm}^2$$

655.48 $${cm}^2$$

Solution

Question 6

4.05 $$\pi m^2$$

5.05 $$\pi m^2$$

6.05 $$\pi m^2$$

7.05 $$\pi m^2$$

Solution

Question 7

188.23 $$m^2$$

168.23 $$m^2$$

178.23 $$m^2$$

198.23 $$m^2$$

Solution

Question 8

3.64 m

3.04 m

3.54 m

3.44 m

Solution

Question 9

560 cm$$^2$$

460 cm$$^2$$

660 cm$$^2$$

760 cm$$^2$$

Solution

Question 10

191.16 m$$^2$$

391.16 m$$^2$$

491.16 m$$^2$$

291.16 m$$^2$$

Solution

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