HOW TO FIND THE VOLUME OF SIMILAR SOLIDS

Problem 1 :

Two soup cans are similar and have heights of 8 cm and 16 cm respectively. Cylinder A has volume 225 cm³.

Find : a) the ratio of their radii.

b) the volume of B.

Solution :

a) When A is enlarged to give B,

k = 16/8 = 2

Therefore the ratio of radii = 1 : 2

b) Volume of  B = k³ × volume of A

= 2³ × 225 cm³

= 1800 cm³

The following contain similar solids. Find the unknown length or volume :

Problem 2 :

Solution :

6 = k × 4

k = 6/4

k = 3/2

Volume of new shape = k³ × volume of old shape

= (3/2)³ × 72

= 27/8 × 72

= 243

So, volume is 243 cm³. 

Problem 3 :

Solution :

large prism = k³ ⋅ small prism

27 = k³ × 8

k³ = 27/8

k = ∛(27/8)

k = 3/2

Length of larger = k ⋅ length of smaller

5 = 3/2 ⋅ x

5 × 2/3 = x

10/3 = x

3 1/3 = x

So, length is 3 1/3 m.

Problem 4 :

Solution :

6 = k × 3

k = 6/3

k = 2

Volume of new shape = k³ × volume of old shape

= k³ × 10

= (2)³ × 10

= 8 × 10

= 80

So, volume is 80 cm³.

Problem 5 :

Solution :

10 = k × 6

k = 10/6

k = 5/3

Volume of new shape = k³ × volume of old shape

= (5/3)³ × 12

= 125/27 × 12

= 55.5

So, volume is 55.5 cm³.

Problem 6 :

Solution :

50 = k³ × 10

k³ = 50/10

k³ = 5

k = ∛5

k = 1.710

Slant height of larger frustum cone = k × Slant height of smaller frustum cone

9 = 1.710 ⋅ x

9/1.710 = x

5.26 = x

So, length is 5.26 cm.

Problem 7 :

Solution :

64 = k³ × 27

k³ = 64/27

k = ∛(64/27)

k = 4/3

Diameter of larger hemisphere = k × diameter of smaller hemisphere

x = 4/3 × 6

x = 8

So, length is 8 cm.

Problem 8 :

The ratio of the corresponding linear measures of two similar solids is 3:5. The volume of the smaller solid is 108 cubic inches. Describe and correct the error in fi nding the volume of the larger solid.

Solution :

Cube of ratio of corresponding sides will be equal to ratio of volume of those solids.

108/V = (3/5)³

108/V = (27/125)

27V = 108(125)

V = 108(125)/27

= 500 cubic inches

So, volume of the required solid is 500 cubic inches.

Problem 9 :

The ratio of the corresponding linear measures of two similar cans of fruit is 4 to 7. The smaller can has a surface area of 220 square centimeters. Find the surface area of the larger can.

Solution :

Ratio between corresponding sides = 4 : 7

Surface area of smaller can = 220 square cm

Surface area of larger can = A

4 : 7 = 220 : A

4/7 = 220/A

4A = 220(7)

A = 220(7)/4

= 385

So, surface area of larger can is 385 sq.cm

Problem 10 :

The volume of a car engine is 390 cubic inches. Which scale model of the car has the greater engine volume, a 1:18 scale model or a 1:24 scale model? How much greater is it?

Solution :

Let A and B be the volumes of car engines which has the ratio of volume 1 : 18 and 1 : 24 respectively.

Volume of car engine = 390 cubic inches

1 : 18 = 390 : A

1/18 = 390/A

A = 390(18)

= 7020

1 : 24 = 390 : A

1/24 = 390/A

A = 390(24)

= 9360

Volume of car engine which has the ratio of 1 : 24 has greater volume

Difference = 9360 - 7020

= 2340 cubic inches

Problem 11 :

You have a small marble statue of Wolfgang Mozart. It is 10 inches tall and weighs 16 pounds. The original statue is 7 feet tall.

a) Estimate the weight of the original statue. Explain your reasoning.

b) If the original statue were 20 feet tall, how much would it weigh?

Solution :

a) Let V be the weight of original statue.

7 : 10 = V : 19

7/10 = V/19

7(19) = 10V

V = 7(19)/10

= 13.3

So, weight of the original statue is 13.3 pounds.

b) 

20 : 10 = V : 19

20/10 = V/19

20(19) = 10V

V = 19(20)/10

= 38

So, weight of the original statue is 38 pounds.