Two prisms are similar. The surface area of one is four times the surface area of the other. What is the ratio of the corresponding dimensions?
side1^2/side^2 = 1/4
side1/side1 = 1/2
the ratio is 1 : 2
To find the ratio of the corresponding dimensions of two similar prisms, we need to compare their surface areas.
Let's assume that the surface area of the smaller prism is represented as S1, and the surface area of the larger prism is represented as S2.
Given that the surface area of one prism is four times the surface area of the other:
S2 = 4 * S1
Since the surface area of a prism is calculated by summing the areas of all its faces, we can express it in terms of the dimensions of the prism.
Let's assume that the dimensions of the smaller prism are represented as l1, w1, and h1, where l1 is the length, w1 is the width, and h1 is the height.
Similarly, let's assume that the dimensions of the larger prism are represented as l2, w2, and h2, where l2 is the length, w2 is the width, and h2 is the height.
The surface area of a prism can be calculated as follows:
S = 2lw + 2lh + 2wh
Now, let's compare the surface areas of the two prisms:
S2 = 2l2w2 + 2l2h2 + 2w2h2
S1 = 2l1w1 + 2l1h1 + 2w1h1
Since we know that S2 is four times S1:
4 * S1 = S2
Substituting the equations for S1 and S2:
4(2l1w1 + 2l1h1 + 2w1h1) = 2l2w2 + 2l2h2 + 2w2h2
Simplifying the equation:
8l1w1 + 8l1h1 + 8w1h1 = 2l2w2 + 2l2h2 + 2w2h2
Dividing both sides of the equation by 2:
4l1w1 + 4l1h1 + 4w1h1 = l2w2 + l2h2 + w2h2
Now, we can compare the corresponding dimensions of the two prisms:
l2w2 : l1w1 = 4:1 (ratio of length multiplied by width)
l2h2 : l1h1 = 4:1 (ratio of length multiplied by height)
w2h2 : w1h1 = 4:1 (ratio of width multiplied by height)
Therefore, the ratio of the corresponding dimensions is 4:1 for all three pairs (length multiplied by width, length multiplied by height, and width multiplied by height).