Surface Areas and Volumes Test

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Surface Areas and Volumes Test - 40

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Weekly Quiz Competition

Question 1
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A vessel is in the form of a frustum of a cone. Its radius at top end is $$10\ inch$$ and the height is $$12\ inch$$. Its volume is $$1200\pi\ {inch}^3$$. Find its bottom radius.

A
$$8.4\ inch$$

B
$$7.5\ inch$$

C
$$7.6\ inch$$

D
$$4.6\ inch$$

Solution

Volume of a frustum cone $$= \displaystyle \frac{\pi h}{3} (R^2 + Rr + r^2)$$
$$\displaystyle 1,200 \pi = \frac{\pi \times 12}{3} (10^2 + 10 \times r + r^2)$$
$$100 \times 3 = (144 + 10r + r^2)$$
$$300 - 144 = 10r + r^2$$
$$r^2 + 10r - 156 = 0$$
On factoring, we get
$$(r - 8.4) (r + 18.45) = 0$$
$$r = 8.4, -18.45$$
Since $$r$$ cannot be negative, thus the bottom radius is $$8.4$$ inches.
Question 2
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The curved surface area of a frustum cone is 34.7 in$$^2$$. The smaller circle area is 15.8 in$$^2$$. The total surface area is 70.5 in$$^2$$. Find the larger circle area of a cone.

A
10 in$$^2$$

B
30 in$$^2$$

C
40 in$$^2$$

D
20 in$$^2$$

Solution

Curved surface area of a frustum cone = $$\pi$$(r + R)s $$\Rightarrow$$  34.7 in$$^2$$
Total surface area of a frustum of cone = $$\pi$$(r + R)s + $$\pi$$r$$^2$$ + $$\pi$$R$$^2$$ $$\Rightarrow$$ 70.5 in$$^2$$
Smaller circle area = $$\pi$$r$$^2$$ $$\Rightarrow$$  15.8 in
Larger circle area = $$\pi$$R$$^2$$ $$\Rightarrow$$  ?
Therefore,

SA = $$\pi$$(r + R)s + $$\pi$$r$$^2$$ + $$\pi$$R$$^2$$
70.5 = 34.7 + Area of larger circle + 15.8
70.5 - 34.7 - 15.8 = Area of larger circle
Area of larger circle of a frustum cone = 20 in$$^2$$
Question 3
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The total surface area of a frustum cone is 1,200 m$$^2$$. The area of larger and smaller circle is 120 and 100 m$$^2$$. Find the curved surface area of a cone.

A
980 m$$^2$$

B
970 m$$^2$$

C
950 m$$^2$$

D
960 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$$^2$$ $$\Rightarrow$$  1,200 m$$^2$$
Smaller circle area = $$\pi$$r$$^2$$ $$\Rightarrow$$  100 m$$^2$$
Larger circle area = $$\pi$$R$$^2$$$$\Rightarrow$$  120 m$$^2$$
Therefore,

SA = $$\pi$$(r + R)s + $$\pi$$r$$^2$$ + $$\pi$$R$$^2$$
1,200 = Curved surface area + 100 + 120
$$1,200 - 100 - 120 =$$ Curved surface area
Curved surface area of a frustum cone = 980 m$$^2$$
Question 4
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The curved surface area of a frustum cone is 250 mm$$^2$$. The smaller circle area is 35 mm$$^2$$. The total surface area is 450 mm$$^2$$. Find the larger circle area of a cone.

A
155 mm$$^2$$

B
165 mm$$^2$$

C
175 mm$$^2$$

D
185 mm$$^2$$

Solution

Curved surface area of a frustum cone = $$\pi$$(r + R)s $$\Rightarrow$$  250 mm$$^2$$
Total surface area of a frustum of cone = $$\pi$$(r + R)s + $$\pi$$r$$^2$$ + $$\pi$$R$$^2$$ $$\Rightarrow$$  450 mm$$^2$$
Smaller circle area = $$\pi$$r$$^2$$ $$\Rightarrow$$  35 mm$$^2$$
Larger circle area = $$\pi$$R$$^2$$ $$\Rightarrow$$  ?
Therefore,
SA = $$\pi$$(r + R)s + $$\pi$$r$$^2$$ + $$\pi$$R$$^2$$
$$450 = 250 $$+ Area of larger circle + 35
$$450 - 250 - 35 = $$Area of larger circle
Area of larger circle of a frustum cone = 165 mm$$^2$$
Question 5
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The curved surface area of a frustum cone is 25$$\pi$$ mm$$^2$$. The larger circle area is 12$$\pi$$ mm$$^2$$. The total surface area is $$350 \pi$$ mm$$^2$$. Find the smaller circle area of a cone.

A
313 mm$$^2$$

B
323$$\pi$$ mm$$^2$$

C
333$$\pi$$ mm$$^2$$

D
313$$\pi$$ mm$$^2$$

Solution

Curved surface area of a frustum cone = $$\pi$$(r + R)s $$\Rightarrow$$ 25$$\pi$$ mm$$^2$$
Total surface area of a frustum of cone = $$\pi$$(r + R)s+ $$\pi$$r$$^2$$ + $$\pi$$R$$^2$$ $$\Rightarrow$$ 350$$\pi$$ mm$$^2$$
Larger circle area = $$\pi$$R$$^2$$ $$\Rightarrow$$ 12$$\pi$$ mm$$^2$$
Smaller circle area = $$\pi$$r$$^2$$ $$\Rightarrow$$ ?
Therefore, SA = $$\pi$$(r + R)s + $$\pi$$r$$^2$$ + $$\pi$$R$$^2$$
350$$\pi$$ = 25$$\pi$$ + Area of smaller circle + 12$$\pi$$
350$$\pi$$ - 25$$\pi$$- 12$$\pi$$ = Area of smaller circle
Area of smaller circle of a frustum cone = 313$$\pi$$ mm$$^2$$
Question 6
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The curved surface area of a frustum cone is 2.34 cm$$^2$$. The larger circle area is 12.5 cm$$^2$$. The total surface area is 25.8 cm$$^2$$. Find the smaller circle area of a cone.

A
12.96 cm

B
11.96 cm

C
10.96 cm

D
9.96 cm

Solution

Curved surface area of a frustum cone = $$\pi$$(r + R)s $$\Rightarrow$$ 2.34 cm$$^2$$
Total surface area of a frustum of cone $$=\pi$$ (r + R)s + $$\pi r^2$$ + $$\pi R^2$$$$\Rightarrow$$ 25.8$$cm^2$$
Larger circle area = $$\pi$$R$$^2$$ $$\Rightarrow$$ 12.5 cm$$^2$$
Smaller circle area = $$\pi$$r$$^2$$ $$\Rightarrow$$ ?
Therefore,
SA = $$\pi$$(r + R)s + $$\pi$$r$$^2$$ + $$\pi$$R$$^2$$
25.8 = 2.34 +Area of smaller circle + 12.5
$$25.8 - 2.34 - 12.5 = $$Area of smaller circle
Area of smaller circle of a frustum cone = 10.96 cm$$^2$$
Question 7
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A vessel is in the form of a frustum of a cone. Its diameter at top end is 2 m and the bottom end is 0.5 m. Its height is 5 m. Find the volume of the frustum cone. (Use $$\pi$$ = 3).

A
1.3125 $$m^3$$

B
2.3125 $$m^3$$

C
6.5625 $$m^3$$

D
4.3125 $$m^3$$

Solution

Volume of a frustum cone = $$ \displaystyle \frac{\pi h}{3} (R^2 + Rr + r^2)$$
Diameter = radius $$\times 2$$
Radius, R = 1 m, r = 0.25 m
= $$ \displaystyle \frac{3 \times 5}{3} (1^2 + 1 \times 0.25 + {0.25}^2)$$
= $$5(1 + 0.25 + 0.0625)$$
= $$5 (1.3125)$$
= 6.5625 $$m^3$$
Question 8
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Find the curved surface area of a frustum cone whose larger and smaller radius is 20 and 5 cm. The slant height is 10 cm. (Use $$\pi$$ = 3.14)

A
765 $$cm^2$$

B
785 $$cm^2$$

C
826$$cm^2$$

D
459 $$cm^2$$

Solution

Curved surface area of a frustum cone = $$\pi$$(r + R)s $$\Rightarrow$$ ?
= 3.14 $$\times$$ (5 + 20) $$\times$$ 10
= 3.14 $$\times$$ (25) $$\times$$ 10
Curved surface area of a frustum cone = 785 cm$$^2$$
Question 9
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A bucket is in the form of a frustum of a cone. The curved surface area of the bucket is $$120 \pi\ {cm}^2.$$ The top and bottom radius of the bucket is $$8\ cm$$ and $$4\ cm.$$ What is the slant height ?

A
$$11\ cm$$

B
$$10\ cm$$

C
$$9\ cm$$

D
$$13\ cm$$

Solution

The curved surface area of a frustum of a cone is given by $$\pi(R+r)l$$ where $$l$$ is the slant height and $$r, R$$ is top and bottom radius.

So, by using the given information,
$$\begin{aligned}{}\pi \left( {R + r} \right)l &= 120\pi \\\left( {8 + 4} \right)l &= 120\\12l &= 120\\l &= 10\ cm\end{aligned}$$

Hence, the slant height of the bucket is $$10\ cm.$$
Question 10
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Find the curved surface area of a frustum cone whose larger and smaller radius is 35 and 10 m. The slant height is 45 m. (Use $$\pi$$ = 3)

A
6,375 m$$^2$$

B
6,275 m$$^2$$

C
6,075 m$$^2$$

D
6,175 m$$^2$$

Solution

Curved surface area of a frustum cone = $$\pi$$(r + R)s $$\Rightarrow$$ ?
= 3 $$\times$$ (35 + 10) $$\times$$45
= 3 $$\times$$ (45) $$\times$$ 45
Curved surface area of a frustum cone = 6,075 m$$^2$$

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Surface Areas and Volumes Test - 40

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Surface Areas and Volumes Test - 40

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Question 1

$$8.4\ inch$$

$$7.5\ inch$$

$$7.6\ inch$$

$$4.6\ inch$$

Solution

Question 2

10 in$$^2$$

30 in$$^2$$

40 in$$^2$$

20 in$$^2$$

Solution

Question 3

980 m$$^2$$

970 m$$^2$$

950 m$$^2$$

960 m$$^2$$

Solution

Question 4

155 mm$$^2$$

165 mm$$^2$$

175 mm$$^2$$

185 mm$$^2$$

Solution

Question 5

313 mm$$^2$$

323$$\pi$$ mm$$^2$$

333$$\pi$$ mm$$^2$$

313$$\pi$$ mm$$^2$$

Solution

Question 6

12.96 cm

11.96 cm

10.96 cm

9.96 cm

Solution

Question 7

1.3125 $$m^3$$

2.3125 $$m^3$$

6.5625 $$m^3$$

4.3125 $$m^3$$

Solution

Question 8

765 $$cm^2$$

785 $$cm^2$$

826$$cm^2$$

459 $$cm^2$$

Solution

Question 9

$$11\ cm$$

$$10\ cm$$

$$9\ cm$$

$$13\ cm$$

Solution

Question 10

6,375 m$$^2$$

6,275 m$$^2$$

6,075 m$$^2$$

6,175 m$$^2$$

Solution

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