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The sides of three cubes are in the ratio five to six to four.
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What is the ratio of their lateral surface areas?
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The keyword here is “areas.”
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A quick way to solve this problem would be to recall that an area scale factor is equal to a length scale factor squared.
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We are told that the side lengths of the cubes are in the ratio five, six, four.
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This means that all three cubes are in proportion to one another and are similar.
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We can therefore find a scale factor that links a cube with side length of one unit to each of the other cubes.
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We calculate the area of a square by squaring its side length.
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This means that the ratio of the areas will be five squared to six squared to four squared.
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These are equal to 25, 36, and 16, respectively.
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The ratio of the lateral surface areas of three cubes whose sides are in the ratio five, six, four is 25, 36, 16.
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A longer way of solving this problem would be to draw the cubes.
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The first cube has side length five units.
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This means that the area of one face would be equal to five squared or five multiplied by five.
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This is equal to 25.
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As a cube has six faces with equal area, the lateral surface area or total surface area will be equal to six multiplied by 25.
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This is equal to 150 square units.
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We can repeat this process for a cube with side length six units.
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This would have a lateral surface area of six multiplied by six squared.
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This is equal to 216 square units.
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Finally, a cube with side length four units would have a lateral surface area of six multiplied by four squared.
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This is equal to 96.
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The ratio of the lateral surface areas is therefore 150, 216, 96.
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All of these values are divisible by six.
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We might have recognized that as all of the cubes had six faces.
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Dividing the three values by six gives us 25, 36, and 16.
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This is the same ratio we found using the first method.