A right triangle prism has a volume of 2 cubic inches. A second right triangle prism is similar to the first one and has a volume of 128 cubic inches.
A. What is the scale factor to go from the first prism to the second?
B. What is the scale factor to go from the second prism to the first?
since the volume scaled by 64, the side length scaled by 4.
1/4 to go the other way.
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To find the scale factor between the two prisms, we can use the formula for volume. The volume of a prism is given by multiplying the base area by the height.
Let's denote the scale factor as "x".
For the first prism:
Volume = 2 cubic inches
For the second prism:
Volume = 128 cubic inches
Using the volume formula, we have:
Volume of the first prism = base area of the first prism * height of the first prism
2 = base area of the first prism * height of the first prism
Similarly, for the second prism, we can write:
Volume of the second prism = base area of the second prism * height of the second prism
128 = base area of the second prism * height of the second prism
A. To find the scale factor to go from the first prism to the second, we need to compare the base areas. Dividing the second equation by the first equation gives:
128 = (base area of the second prism / base area of the first prism) * (height of the second prism / height of the first prism)
Simplifying the equation, we get:
128 = (x^2) * (x)
128 = x^3
Taking the cube root of both sides, we find:
x = ∛(128)
x ≈ 4
Therefore, the scale factor to go from the first prism to the second is approximately 4.
B. To find the scale factor to go from the second prism to the first, we need to compare the base areas again. Dividing the first equation by the second equation gives:
2 = (base area of the first prism / base area of the second prism) * (height of the first prism / height of the second prism)
Simplifying the equation, we get:
2 = (1 / x^2) * (1 / x)
2 = 1 / (x^3)
Rearranging the equation, we find:
x^3 = 1/2
Taking the cube root of both sides, we get:
x = ∛(1/2)
x ≈ 0.7937
Therefore, the scale factor to go from the second prism to the first is approximately 0.7937.
To solve this problem, we need to understand the relationship between the volumes of similar 3D shapes. The volumes of similar shapes are related by the cube of the scale factor.
Let's first find the scale factor to go from the first prism to the second:
Step 1: Find the cube root of the volumes ratio:
Cube root of (Second volume / First volume) = Cube root of (128 / 2) = Cube root of 64 = 4
Step 2: The scale factor from the first prism to the second prism is the result from the above step, which is 4.
Now, let's find the scale factor to go from the second prism to the first:
Step 1: Find the cube root of the volumes ratio:
Cube root of (First volume / Second volume) = Cube root of (2 / 128) = Cube root of (1 / 64) = 1/4
Step 2: The scale factor from the second prism to the first prism is the result from the above step, which is 1/4.
So, the answers to the questions are:
A. The scale factor to go from the first prism to the second is 4.
B. The scale factor to go from the second prism to the first is 1/4.