We have a triangular prism that has the following characteristics. It has a volume of 360 cubic inches. The base of the triangle is twice the height of the triangle. The difference between the height of the prism and the height of the triangle is 4 inches. The height of the prism is 10 inches. Need help finding the dimensions of this triangular prism.

Is the triangular cross section equilateral? It is for most triangular prisms.

If the triangle is equilateral, it seems to me they have provided too much information. If it is scalene, there is not enough information.

You know that the length or "height" of the prism is 10 inches and the triangle height is 6 inches (or 14 inches). The triangle cross section is 360/10 = 36 square inches. The base of the triangle is
2(Area)/(triangle height) = 72/6 = 16 in.

To find the dimensions of the triangular prism, let's break down the problem step by step:

Step 1: Determine the dimensions of the triangular base.
The base of the triangle is twice the height of the triangle. We can represent the base as 2h, where h is the height of the triangle.

Step 2: Find the height of the triangle.
The difference between the height of the prism and the height of the triangle is 4 inches. Given that the height of the prism is 10 inches, we can represent the height of the triangle as h = 10 - 4 = 6 inches.

Step 3: Calculate the area of the base triangle.
The area of a triangle can be calculated using the formula: A = (1/2) * base * height. In our case, the base is 2h and the height is h.

A = (1/2) * (2h) * h
= h^2

Step 4: Find the dimensions of the triangular base.
Since we know the volume of the triangular prism is 360 cubic inches, we can use the formula: volume = base area * height.

360 = A * height
= h^2 * height

Step 5: Substitute the known values into the equation.
Replacing A with h^2 and height with 10 (height of the prism), we get:

360 = h^2 * 10
36 = h^2

Step 6: Solve for h.
Take the square root of both sides of the equation:

√36 = √(h^2)
6 = h

Therefore, the height of the triangle, h, is 6 inches.

Step 7: Calculate the base dimensions.
The base is 2h, so the base dimensions are:

base = 2 * 6
= 12 inches

Step 8: Write the final dimensions.
Given that the height of the prism is 10 inches, we have:

Height = 10 inches
Base = 12 inches
Triangle Height = 6 inches

So, the dimensions of the triangular prism are 10 inches (height), 12 inches (base), and 6 inches (triangle height).