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Similar Solids Worksheet - Page 3

Similar Solids Worksheet
  • Page 3
 21.  
If the ratio of surface areas of two similar solids is less than 1, then what is the similarity ratio of the solids?
a.
= 1
b.
> 1
c.
> 2
d.
< 1


Solution:

Let the ratio of surface areas of two similar solids = 0.09.
[Ratio of surface areas < 1.]

Let the similarity ratio of these solids = k.

k2 = 0.09
[Definition.]

k = 0.3
[Find the square root of each side.]

Clearly, k = 0.3 < 1.


Correct answer : (4)
 22.  
The similarity ratio of two similar solids is k, which is less than 1. The ratio of their surface areas is l and the ratio of their volumes is m. Which of the following is correct?
a.
l < k < m
b.
m < l < k
c.
l < m < k
d.
k < l < m


Solution:

Let k = 0.2
[k < 1.]

Ratio of the surface area l = k2
[Definition.]

= (0.2)2 = 0.04
[Substitute 0.2 for k.]

Ratio of volumes = m = k3
[Definition.]

= (0.2)3 = 0.008
[Substitute 0.2 for k.]

So k < l < m.


Correct answer : (4)
 23.  
If the similarity ratio, ratio of surface areas and the ratio of volumes of two similar solids are l, m and n respectively, then which of the following is correct?
a.
m = ln
b.
m2 = ln
c.
l2 = mn
d.
n2 = lm


Solution:

Similarity ratio of the two similar solids = l
[Given.]

Ratio of the surface area = m
[Given.]

Ratio of the volume = n
[Given.]

l2 = m and l3 = n
[Definition.]

So, m2 = (l2)2 = l4.
[Substitute l2 for m.]

l4 = l (l3) = ln
[Substituting n for l3.]

So, m2 = ln.


Correct answer : (2)
 24.  
If the similarity ratio and the ratio of surface areas of two similar solids are 2x and x16, then what is the ratio of the volumes of the solids?
a.
4 : 1
b.
2 : 1
c.
1 : 4
d.
1 : 2


Solution:

Similarity ratio of the two solids = 2x
[Given.]

Ratio of surface area of solids = x16
[Given.]

(2x)2 = x16
[Definition.]

4x2=x16

x3 = 64
[Cross - product property.]

x = 4
[Find cube root on each side.]

So, the similarity ratio = 2x=24=12.
[Substitute 4 for x.]

So, the ratio of the surface areas = (1 / 2)2
[Definition.]

= 1 / 4 = 1 : 4


Correct answer : (3)
 25.  
The similarity ratio, ratio of surface areas and ratio of volumes of two similar solids are 2l, 4m and 8n respectively. Which of the following is correct?
a.
l3 = mn
b.
l5 = mn
c.
l4 = mn
d.
l2 = mn


Solution:

Similarity ratio of the two solids = 2l
[Given.]

Ratio of surface areas of the solids = 4m
[Given.]

Ratio of volumes of the solids = 8n
[Given.]

(2l)2=4m and (2l)3=8n
[Definition.]

4l2=4m and 8l3=8n

l2 = m and l3 = n
[Cross - product property.]

l5 = (l2)(l3) = mn
[Substitute l2 for m and l3 for n.]

So, l5 = mn.


Correct answer : (2)

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