Maiia B. answered • 03/11/17

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^{3}: 3

^{3}, or 8:27.

Lydia S.

asked • 03/11/17The surface areas of two similar shapes are in the ratio 4:9. What is the ratio of their volumes?

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Maiia B. answered • 03/11/17

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3
(6)
Mathematician with degree, vast knowledge and patience

John M. answered • 03/11/17

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Engineering manager professional, proficient in all levels of Math

- Since the ratio of their two-dimensional property (i.e., surface area) is 4:9, we can take the square root of this to determine the relationship of their one dimension.
- Sqrt(4/9) = Sqrt(4) / Sqrt(9) = 2/3
- Volume is a three dimensional property, so the ratio of the two shapes is
**(2/3)**^{3}= 8/27.

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Here's another way to arrive at the same answer, using a cube as the shape.

- The surface area of a cube is just 6* area of one surface. The reason for multiplying by 6 is that there are 6 surfaces on a cube. The area of a surface is the length of a side squared. Putting it altogether, the Surface Area (SA) of a cube = 6* s
^{2} - If there are two cubes, C1 and C2, that have a SA ratio of 4:9, then 9*SA
_{C1}= 4*SA_{C2 }{Eqn 1} - Let s1 = length of each side of C1. And s2 = length of each side of C2. Then Eqn 1 can be rewritten as:
- 9*(6s
_{1}^{2}) = 4*(6s_{2}^{2}) - 54s
_{1}^{2}= 24s_{2}^{2} - 54/24s
_{1}^{2}= s_{2}^{2} - sqrt(54/24s
_{1}^{2}) = s_{2} - s
_{2}= sqrt(54/24) * sqrt(s_{1}^{2}) - s
_{2}= sqrt(54/24) * s_{1 } - s2 = (54/24)
^{1/2}* s_{1}{Eqn 2} - The volume V of a cube is simply s
^{3} - V
_{C1}= s_{1}^{3} - V
_{C2}= s_{2}^{3}{Eqn 3} - Substitute Eqn 2 into Eqn 3. V
_{C2}= (54/24)^{1/2}* s_{1})^{3} - V
_{C2}= (54/24)^{3/2}* s_{1}^{3} - V
_{C2}= 3.375s_{1}^{3} - Now that the volumes are both expressed as a function of s
_{1}, we can compare their ratio - V
_{C1}/V_{C2}= s_{1}^{3}/ 3.375s_{1}^{3} - V
_{C1}/V_{C2}=**0.296296** - If you want to express this as an integer ratio:
**8: 27.**

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