# 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.

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.

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. $m$ = $\mathrm{ln}$ b. $m$2 = $\mathrm{ln}$ c. $l$2 = $\mathrm{mn}$ d. $n$2 = $\mathrm{lm}$

#### 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.

24.
If the similarity ratio and the ratio of surface areas of two similar solids are $\frac{2}{x}$ and $\frac{x}{16}$, 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

25.
The similarity ratio, ratio of surface areas and ratio of volumes of two similar solids are $\frac{2}{l}$, $\frac{4}{m}$ and $\frac{8}{n}$ respectively. Which of the following is correct?
 a. $l$3 = $\mathrm{mn}$ b. $l$5 = $\mathrm{mn}$ c. $l$4 = $\mathrm{mn}$ d. $l$2 = $\mathrm{mn}$

#### 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.