Similar Solids Worksheet - Page 2

Similar Solids Worksheet
• Page 2
11.
Which of the following is / are correct?
1. Similar solids have same shape.
2. The ratio of surface areas of similar solids is the same as the ratio of its volumes.
3. The ratio of volumes of two similar solids is thrice the ratio any of its corresponding sides.
4. Similar solids have same volume.
 a. (1) and (2) only b. (1) and (4) only c. (1) only d. (1) and (3) only

Solution:

Similar solids have same shape and all their corresponding dimensions are proportional.

So similar solids do not possess the same dimensions.

Ratio of surface areas of similar solids is the square of the similarity ratio.

Ratio of volumes of similar solids is the cube of its similarity ratio.

So, statement (1) only is correct.

12.
A wooden paperweight in the shape of a cone weighs 0.12 lb. What is the weight of a similar paperweight, which has double the dimensions of the first one?
 a. 0.96 lb b. 0.6 lb c. 0.5 lb d. 2 lb

Solution:

The similarity ratio is 1 : 2.

So, the ratio of the volume is 13 : 23 or 1 : 8.

1 / 8 = 0.12x
[Let x be the weight of the larger paper weight.]

x = 0.96
[Cross-product property.]

The larger paperweight weighs 0.96 lb.

13.
A rectangular matchbox contains 45 sticks. If a jumbo box is made by doubling all the dimensions of this rectangular matchbox, then how many sticks can the jumbo box hold?
 a. 180 b. 2025 c. 90 d. 360

Solution:

The similarity ratio is 1 : 2.

So, the ratio of the volumes is 13 : 23 or 1 : 8.

1 / 8 = 45x
[Let the number of sticks in the jumbo box be x.]

x = 360
[Cross - product property.]

The number of sticks that jumbo matchbox can hold = 360.

14.
The similarity ratio and the ratio of volumes of two similar solids are $x$ : 2 and 8 : $x$ . Find the ratio of their surface areas.
 a. 2 : 1 b. 4 : 1 c. 8 : 1 d. 16 : 1

Solution:

Similarity ratio of the two similar solids = x2

Ratio of volumes of the two solids = 8x

(x2)3=8x
[Definition.]

x4 = 256
[Cross - product property.]

x = 4
[Find fourth root of each side.]

So, similarity ratio = x2=42 = 2 :1
[Substitute 4 for x.]

So, the ratio of the surface areas of the two solids = (2)2 : (1)2 = 4 : 1
[Definition of ratio of surface areas.]

15.
Which of the following is correct?

 a. B b. A c. D d. C

Solution:

ab=12
[Similarity ratio is a : b.]

a2b2=(ab)2=(12)2=14
[Substitute 1 / 2 for ab.]

So, ratio of the surface areas is a3 : b3 = 1 : 8
[Substitute 1 / 2 for ab.]

So, ratio of the volumes is a3 : b3 = 1 : 8.
[Substitute 1 / 2 for ab.]

16.
The cones shown in the figure are similar. Find the value of $l$ if $a$ = 6 units, $b$ = 9 units and $x$ = 8 units.

 a. 17 units b. 15 units c. 12 units d. 14 units

Solution:

Since the corresponding dimensions of two similar solids are proportional, 68=9l

3 / 4 = 9l

l = 12
[Cross - product property.]

17.
The cones shown in the figure are similar. Find the area of the base of smaller cone if $l$ = 6 units, $x$ = 10 units, and $y$ = 12 units.

 a. 400$\pi$ units2 b. 100$\pi$ units2 c. 25$\pi$ units2 d. 5$\pi$ units2

Solution:

The corresponding dimensions of the two similar cones shown are proportional, 612 =r10

1 / 2 = r10

r = 5

The area of the base of the smaller cone = πr2

= π(5)2 = 25π units2
[Substitute 3 for r.]

18.
The cylinders shown are similar. Find the volume of the larger cylinder if $x$ = 8 units, $a$ = 12 units and $b$ = 15 units.

 a. 1500 $\pi$ units3 b. 768 $\pi$ units3 c. 960 $\pi$ units3 d. 300 $\pi$ units3

Solution:

The corresponding dimensions of the two similar cylinders shown are proportional, 1215=8r

4 / 5 = 8r

r = 10

The volume of the larger cylinder = πr2h
[Formula.]

= π × 102 × 15 = 1500 π units3
[Substitute 10 for r and 15 for h.]

19.
The rectangular boxes shown are similar. The volume of the larger box is _______ .
[Given $a$ = 3 units, $b$ = 4 units, $c$ = 5 units and $x$ = 9 units.]

 a. 15 cubic units b. 1620 cubic units c. 3240 cubic units d. 810 cubic units

Solution:

Since the corresponding dimensions of the two similar boxes shown are proportional, 39=4y and 39=5h.

1 / 3= 4y and 1 / 3= 5h

y = 12 and h = 15
[Cross - product property.]

The volume of the larger box = xyh.

= 9 × 12 × 15 = 1620 cubic units
[Substitute 12 for b and 15 for h.]

20.
The similarity ratio, ratio of surface areas and of volumes of two similar solids are in
 a. no specific relation b. arithmetic progression c. harmonic progression d. geometric progression

Solution:

Let the similarity ratio of two similar solids be k.

So, the ratio of their surface areas = k2 and the ratio of their volumes = k3.
[Definition.]

Clearly k, k2, k3 are in geometric progression.