﻿ Volume of a Prism Worksheet | Problems & Solutions # Volume of a Prism Worksheet

Volume of a Prism Worksheet
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1.
A circular hole of diameter $d$ = 3 m is digged on a levelled plot of land shown to a depth of 4 m and the sand so obtained is spread evenly over the rectangular portion of measurement $a$ × $b$ = 7 m × 14 m. What is the increase in the level of the rectangular plot? [Take $\pi$ = 3.]  a. 36.55 cm b. 27.55 cm c. 29.55 cm d. 33.55 cm

#### Solution:

Increase in the level = Volume of the sand Surface area
[Formula.]

Radius = d / 2= 1.50 m = 150 cm

Height = 4 m = 400 cm.
[Given.]

Volume of the sand = Volume of the hole = 3 × 1502 × 400 = 27000000 cm3
[Volume of cylinder = π r2 h.]

Area of the rectangular plot = 700 × 1400 = 980000 cm2
[Area = a × b.]

Increase in the level of the plot = 27000000 / 980000 = 27.55 cm
[Substitute in step 1 and simplify.]

2.
A circular tank of thickness 30 cm and height 80 cm whose diameter is 4 m is to be constructed in a storage yard. The rate of construction is $10 per m3. Find the cost of construction. [Take $\pi$ = 3.] a.$14.94 b. $3090 c.$284 d. $30.90 #### Solution: Inner radius of the tank = d / 2= 2 m = 200 cm [Radius = Diameter / 2.] Outer radius of the tank = 200 + 30 = 230 cm [Radius + thickness of the tank.] Height of the tank = 80 cm [Given.] Volume of the tank = Volume of outer cylinder - Volume of the inner cylinder [Formula.] Volume of the outer cylinder = 3 × (230)2 × 80 = 12696000 cm3 [Volume of the cylinder = π r2 h.] Volume of the inner cylinder = 3 × (200)2 × 80 = 9600000 cm3 [Volume of the cylinder = π r2 h.] Volume of the tank = 12696000 - 9600000 = 3096000 cm3 = 3.09 m3 [Substitute in step 4.] Rate of construction =$10 per m3
[Given.]

Cost of construction = 3.09 m3 × 10 = \$30.90
[Simplify.]

3.
A golden bar of length 1.40 m and cross section 18 cm × 18 cm is used to make circular chains of length 64 cm and diameter 6 mm. How many such chains can be made? [Take $\pi$ = 3.] a. 2668 b. 2625 c. 2659 d. 2811

#### Solution:

Number of chains that can be made = Volume of the golden bar Volume of one chain
[Formula.]

Volume of the golden bar = 1.40 × 100 ×18 × 18 = 45360 cm3
[Convert meter to centimeter.]

Radius of the chain = d / 2= 3 mm = 0.3 cm

Volume of one chain = 3 × (0.3)2 × 64 = 17.28 cm3
[Volume of a cylinder = π r² h.]

Number of chains = 45360 / 17.28= 2625 chains (approximately)
[From steps 2 and 4.]

4.
A medicine is prepared in a tank of side length 1.4 m, width 1 m and depth 3 m. The medicine is filled in cylindrical bottles of diameter 8 cm and height 18 cm. How many bottles can be filled? [Take $\pi$ = 3.] a. 10 b. 4861 c. 49 d. 1215

#### Solution:

Number of bottles = Volume of the tank Volume of one bottle
[Formula]

Radius of the bottle = d / 2= 4 cm

Volume of the bottle = 3 × 42 × 18 = 864 cm3
[Volume of a cylinder = π r² h.]

Volume of the tank = 1.4 × 1 × 3 m³ = 4.2 m³ = 4200000 cm3
[Convert meters to centimeters.]

Number bottles = 4200000 / 864= 4861 (approximately)
[From steps 3 and 4.]

5.
A circular wall of radius 2 m and thickness 30 cm is to be constructed to a total height of 0.8 m. Bricks of measurement 15 cm × 8 cm × 8 cm are to be used. How many bricks would be needed? [Take $\pi$ = 3.] a. 3281 b. 3312 c. 3248 d. 3225

#### Solution:

Number of bricks = Volume of the wall Volume of one brick
[Formula.]

Volume of the wall = 3[((2 × 100) + 30)2 - (2 × 100)2]80 = 3096000 cm3
[Volume of ring = π ((R+t)2 - R2) h.]

Volume of the brick = 15 × 8 × 8 = 960 cm3
[Volume of the brick = a × b × c.]

Number of bricks = 3096000 / 960 = 3225 (approximately)
[From steps 2 and 3.]

6.
A 2 m long cylindrical drum of diameter 1m is lying on its side. A dip-stick is inserted which shows a 30 cm depth of liquid. What volume of liquid is contained in the cylinder? a. 0.396 m3 b. 0.208 m3 c. 0.5652 m3 d. 1.317 m3

#### Solution: Volume of the liquid = Area of the segment ACB × Length of the drum
[Formula.]

Area of the segment ACB= Area of the sector AOB - Area of the triangle AOB
[From the figure.]

The depth of the water CD is 30 cm
[Given.]

Then OD = d / 2- y = 100 / 2- 30 = 20 cm
[From the figure.]

cos AOD = OD / AO= 20 / 50
[From the figure.]

cos AOD = 0.4 = AOD = cos-1 0.4 = 66.42°
[Substitute the values.]

Let measure of AOD be θ

Then, AOB = 2 AOD = 2θ

Area of the sector AOB = 2θ360 × π ×d²4
[Formula.]

Area of the sector AOB = 132.84 / 360× 3.14 × 100² / 4= 2896.65 cm2
[Substitute the values and simplify.]

Area of the triangle AOB = 2 × 1 / 2× AO × OD × sin θ
[Since AD = AO sin θ.]

Area of the triangle AOB = 2 × 1 / 2× 50 × 20 × 0.9165 = 916.5 cm2
[Substitute the values: Radius (AO) = 50 cm, OD = 20 cm.]

Area of the segment ACB = 2896.65 - 916.5 = 1980.15 cm 2
[Substitute in step 2 and simplify.]

Length of the drum = 200 cm
[Given.]

Volume of the liquid = 1980.15 × 200 = 396030 cm3 or 0.396 m3
[Substitute in step 1 and simplify.]

7.
The volume of a rectangular prism is 300 in3. If the width of the prism is 5 in and the height is 6 in, then find the length of the rectangular prism. a. 20 in b. 27 in c. 15 in d. 10 in

#### Solution:

Volume of a prism = length × width × height.

V = l × w × h
[Write an equation.]

l = Vw × h

= 3006 × 5
[Substitute V = 300, w = 5 and h = 6.]

= 300 / 30
[Multiply 5 and 6.]

= 10 in
[Divide 300 by 30.]

The length of the prism is 10 in.

8.
The volume of a rectangular prism is 567 cm3. If the width of the prism is 9 cm and the height is 9 cm, what is the length of the rectangular prism? a. 9 cm b. 8 cm c. 10 cm d. 7 cm

#### Solution:

The volume of the prism = length × width × height

= 5679 × 9
The length of the prism = volumewidth × height
[Substitute the values.]

= 56781
[Multiply 9 by 9.]

= 7 cm
[Divide.]

The length of the prism is 7 cm.

9.
If the length, width and the height of a rectangular prism are increased 3 times, then the volume of the prism increases _______________. a. 28 times b. 23 times c. 27 times d. 18 times

#### Solution:

Let l, w and h be the length, width and height of the prism, respectively.

The volume of the prism = l × w × h

The new length of the prism = 3l
[Since length is increased by 3 times.]

The new width of a prism = 3w
[Since width is increased by 3 times.]

The new height of the prism = 3h
[Since height is increased by 3 times.]

= 3l × 3w × 3h = 27(lwh)
The new volume of a prism = length × width × height
[Multiply.]

The volume of the prism increases by 27 times.

10.
Five copper cubes of sides 8 cm, 5 cm, 4 cm, 4 cm and 1cm are melted to make a single cube. What is the volume of the new cube so formed? a. 866 cm3 b. 766 cm3 c. 22 cm3 d. 756 cm3

#### Solution:

Volume of the new cube, V = Sum of the volumes of all five cubes.

= 83 + 53 + 43 + 43 + 13
[Volume of a cube = (side)3.]

= 512 + 125 + 64 + 64 + 1
[Substitute the values and simplify.]

= 766

Volume of the new cube, V = 766 cm3