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Similar Solids Worksheet

Similar Solids Worksheet
  • Page 1
 1.  
Two similar cylinders have heights 3 cm and 2 cm. What is the similarity ratio?.
a.
3 : 2
b.
4 : 3
c.
3 : 4
d.
2 : 3


Solution:

Heights of the two similar cylinders are a = 3 cm and b = 2 cm.
[Given.]

The ratio of corresponding dimensions of two similar solids is the similarity ratio.
[Definition.]

Similarity ratio = Height of the first cylinderHeight of the second cylinder

a : b = 3 : 2
[Substitute 3 for a and 2 for b.]


Correct answer : (1)
 2.  
Two similar cones have heights 8 cm and 4 cm. What is the ratio of their surface areas?
a.
2 : 1
b.
6 : 1
c.
8 : 1
d.
4 : 1


Solution:

Heights of the two similar cylinders are a = 8 cm and b = 4 cm.
[Given.]

The ratio of corresponding dimensions of two similar solids is the similarity ratio.
[Definition.]

Similarity ratio = Height of the first coneHeight of the second cone

= ab = 8 / 4 = 2 : 1
[Substitute 8 for a and 4 for b.]

The ratio of their corresponding surface area is a2 : b2.
[Theorem.]

= (ab)² = 4 : 1
[Substitute and simplify.]


Correct answer : (4)
 3.  
The radii of two spheres are 3 cm and 2 cm. Find the ratio of their volumes approximately.
a.
9 : 4
b.
27 : 8
c.
3 : 2
d.
2 : 3


Solution:

The radii of the two spheres are a = 3 cm and b = 2 cm.
[Given.]

Similarity ratio = Radius of first sphereRadius of second sphere.
[Formula.]

= ab = 3 / 2
[Substitute 3 for a and 2 for b.]

The ratio of Volumes of the solids is a3 : b3 = 27 : 8 .
[Formula.]


Correct answer : (2)
 4.  
The ratio of areas of two similar solids is 9 : 1. What is the similarity ratio of the solids?
a.
3 : 1
b.
9 : 1
c.
1 : 3
d.
1 : 9


Solution:

a²b² = 91
[The ratio of the areas is a2 : b2.]

ab = 31
[Find the square root of each side.]

So, the similarity ratio of the two solids is a : b = 3 : 1


Correct answer : (1)
 5.  
The ratio of volumes of two similar solids is 27 : 8. What is the similarity ratio of the solids?
a.
3 : 2
b.
9 : 4
c.
4 : 9
d.
2 : 3


Solution:

a3b3 = 278
[The ratios of the volumes is a3 : b3.]

ab = 32
[Find the cube root of each side.]

So, the similarity ratio of the two solids is a : b = 3 : 2 .


Correct answer : (1)
 6.  
The ratio of the base areas of two similar cylinders is 1 : 4. What is the ratio of the volumes of the solids?
a.
1 : 2
b.
8 : 1
c.
1 : 4
d.
1 : 8


Solution:

a2b2 = 1 / 4
[The ratios of the base areas is a2 : b2.]

ab = 1 / 2
[Find the square root on each side.]

So, a3b3 = (ab)3 = (12)3 = 1 / 8
[Substitute 1 / 2 for ab.]

The ratio of volumes of the two solids is a3 : b3 = 1 : 8.


Correct answer : (4)
 7.  
The ratio of volume of two similar cones is 1 : 512. Find the ratio of the base areas of the two solids.
a.
64 : 1
b.
1 : 64
c.
1 : 8
d.
8 : 1


Solution:

a3b3 = 1 / 512
[The ratios of the volume is a3 : b3.]

ab = 1 / 8
[Find the cube root of each side.]

So, a2b2=(ab)2=(18)2 = 164.
[Substitute (1 / 8) for ab.]

The ratio of base areas of the two solids is a2 : b2 = 1 : 64.


Correct answer : (2)
 8.  
The ratio of the volumes of two similar solids is 8 : 27. What is the volume of the smallest one?
a.
3 cubic units
b.
2 cubic units
c.
8 cubic units
d.
cannot be determined


Solution:

The ratio of the volumes is 8 : 27.
[Given.]

a3b3=827
[The ratio of the volume is a3 : b3.]

Without knowing the volume of the larger solid b3, the volume of the smaller solid cannot be determined.


Correct answer : (4)
 9.  
The heights of two similar solids are 4 cm and x cm. The similarity ratio of these two similar solids is 2 : 3. Find the value of x.
a.
4
b.
7
c.
6
d.
8


Solution:

ab=23
[Similarity ratio is a : b.]

4x =23
[Substitute 4for a and x for b.]

x = 6
[Cross product property.]


Correct answer : (3)
 10.  
The surface area of two similar spheres are 144π cm2 and 16π cm2. The volume of the smaller solid is 32 3π cm3. What is the volume of the larger sphere?
a.
144 π cm3
b.
288 π cm3
c.
288 cm3
d.
144 cm3


Solution:

a2b2=144π16π
[The ratio of the surface areas is a2 : b2.]

a2b2 = 9
[Divide numerator and denominator by the common factor 16π.]

ab = 3
[Find the square root of each side.]

Let v1, v2 be the volumes of the two similar spheres.

v1v2=33 v1 = v2(27)
[The ratios of the volumes is a3 : b3.]

v1 = 32 / 3π(27)
[Substitute 32 / 3π for v2.]

v1 = 288 π
[Cross product property.]

The volume of the larger sphere is 288 π cm3.


Correct answer : (2)

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