A fuel storage tank is in the shape of an equilateral triangular prism. The prism is 8 feet high and 10 feet long. When the tank is half-full, the depth of the fuel is what?
note: I know the only way to find the depth is to cut the triangle in half horizontally creating another triangle and a trapezoid and make the height of the trapezoid the depth of the half full tank (variable h) but how do i find that?/?
To find the depth of the half-full fuel in the tank, you can use the concept of similar triangles. Here's how you can approach it:
1. First, visualize the equilateral triangular prism. It has a height of 8 feet, which means the vertical side of the equilateral triangle is also 8 feet.
2. Now, to determine the depth of the half-full tank, imagine cutting the prism horizontally at half of its height, which is 4 feet. This cut creates a smaller equilateral triangle and a trapezoid.
3. Let's focus on the smaller equilateral triangle. Since it is similar to the larger equilateral triangle, it has proportional sides. Therefore, the vertical side of the smaller triangle is also 4 feet.
4. Now, you have a new triangle with two known sides: the vertical side (4 feet) and the horizontal side (half of the original side). The length of the original side of the triangle is 10 feet, so half of that is 5 feet.
5. Using the Pythagorean theorem, we can find the length of the base of the smaller equilateral triangle. Let's call it 'x.'
x^2 = (10/2)^2 - 4^2
x^2 = 5^2 - 4^2
x^2 = 25 - 16
x^2 = 9
x = 3 feet
Thus, the base of the smaller equilateral triangle is 3 feet.
6. Now, you can calculate the depth of the half-full tank by subtracting the base of the smaller triangle from the height of the trapezoid. Let's call it 'h.'
h = 8 feet - 3 feet
h = 5 feet
Therefore, the depth of the fuel when the tank is half-full is 5 feet.
think about similar triangles.
1/2 the altitude means 1/4 the area
or, conversely, 1/2 the area means 1/√2 the altitude