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Math Chapter 9 : Topic 2
Area and Volume Ratios
1 / 20
If two similar triangles have sides in the ratio $3\ :5$, then the ratio of their areas is:
Reasoning: Area ratio $=$ square of sides ratio $\Longrightarrow$ $3^{2}\ :5^{2} = 9\ :25$The surface area of similar solids is proportional to the square of the ratio of their corresponding lengths.
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If the areas of two similar figures are in the ratio $16\ :25$, then the ratio of their corresponding sides is:
Reasoning: $Side\ ratio = \ \ \ \ \Longrightarrow \ \ \ \ \ \ : = 4\ :5$The surface area of similar solids is proportional to the square of the ratio of their corresponding lengths.
3 / 20
Two cubes are similar. If their edges are in the ratio $2\ :3$, then the ratio of their volumes is:
Reasoning: Volume ratio $=$ cube of sides ratio $\Longrightarrow$ $2^{3}\ :3^{3} = 8\ :27$The volume of similar solids is proportional to the cube of the ratio of their corresponding lengths.
4 / 20
If the volume of a sphere is increased by a factor of $8$, the radius is multiplied by:
Reasoning: Volume $\propto r^{3}$, so $\sqrt[{3\ }]{8}\ = 2$
5 / 20
Two similar cylinders have heights in the ratio $1\ :3$. Their volumes are in the ratio:
Reasoning: Volume ratio $=$ cube of linear ratio $\Longrightarrow$ $1^{3}\ :3^{3} = 1\ :27$
6 / 20
If two similar figures have side ratio $5\ :7$, what is the ratio of their areas?
Reasoning: Area ∝ square of sides $\Longrightarrow$ $5^{2}\ :7^{2} = 25\ :49$
7 / 20
The areas of two similar figures differ by a factor of $36$. The scale factor between them is:
Reasoning: Scale factor $= = 6$
8 / 20
If the radius of a sphere is doubled, its volume becomes:
Reasoning: Volume $\propto r^{3}$, so $2^{3} = 8$
9 / 20
Two similar solids have volumes in ratio $64\ :125$. Their corresponding sides are in ratio:
Reasoning: Side ratio $=$ cube root $\Longrightarrow$ $4\ :5$
10 / 20
If areas of two squares are in ratio $49\ :81$, then their side lengths are in ratio:
Reasoning: Side ratio $= \ \ \ \ \Longrightarrow \ \ \ \ 7\ :9$
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If the volume of a cube is increased by $27$ times, its edge becomes:
Reasoning: Cube root of $27 = \sqrt[{3\ }]{27} = 3$
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Two similar cones have radii in ratio $3\ :4$. The ratio of their volumes is:
Reasoning: Volume $\propto$ cube $\Longrightarrow$ $3^{3}\ :4^{3} = 27\ :64$The volume of similar solids is proportional to the cube of the ratio of their corresponding lengths.
13 / 20
If two similar figures have areas in ratio $1\ :100$, the scale factor is:
Reasoning: $= 10\ \ \ \ \Longrightarrow \ \ \ \ Side\ ratio = \ 1\ :10$
14 / 20
If the dimensions of a solid are tripled, its volume increases by:
Reasoning: Volume $\propto$ cube $\Longrightarrow$ $3^{3} = 27$
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Two similar solids have surface areas in ratio $36\ :49$. Their volumes are in ratio:
Reasoning: Side ratio $= 6\ :7\ \ \ \ \Longrightarrow \ \ \ \ $Volume ratio $= 6^{3}\ :7^{3} = 216\ :343$
16 / 20
If the area of a figure is increased by $9$ times, its side length is multiplied by:
Reasoning: $= 3$
17 / 20
If two similar spheres have volumes $8\ {cm}^{3}$ and $64\ {cm}^{3}$, their radii ratio is:
Reasoning: Cube root of $8\ :$ $64\ \ \ \ \Longrightarrow \ \ \ \ 2\ :4 = 1\ :2$
18 / 20
If side ratio is $1\ :4$, the area ratio is:
Reasoning: Square of ratio $\Longrightarrow$ $1^{2}\ :4^{2} = 1\ :16$
19 / 20
If side ratio is $1\ :4$, the volume ratio is:
Reasoning: Cube of ratio $\Longrightarrow$ $1^{3}\ :4^{3} = 1\ :64$
20 / 20
A cube and a similar cube have volumes $125\ {cm}^{3}$ and $1000\ {cm}^{3}$. The ratio of edges is:
Reasoning: Cube root $\Longrightarrow$ $5\ :10\ \ = 1\ :2$
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