To cut down on the cost of product, a laundry detergent company decides to make the inside of their rectangular cardboard box into a pyramid. If the original volume held 288 in.3 of detergent, how much does the new design hold?
To find the volume of a pyramid, you use the formula V = 1/3 * B * h, where B is the area of the base and h is the height of the pyramid.
Since the original box is rectangular, the base is a rectangle. Since the new design is a pyramid, the base is now a triangle. We must find the dimensions of this triangle.
Let's say the original dimensions of the rectangular box are length = L, width = W, and height = H.
The volume of the original rectangular box is L * W * H = 288 in.^3.
Now, since the inside of the box is now a pyramid, the base of the pyramid will be a triangle with base B = W and height h = H. The volume of the pyramid is now V = 1/3 * W * H * H = 1/3 * W * H^2.
Since the original volume is the same as the new volume, we can set the equations equal to each other:
L * W * H = 1/3 * W * H^2
Now, we can solve for the new height of the pyramid (H) in terms of the original height (H):
3L = H
Now, we can substitute this into the formula for the volume of the pyramid:
V = 1/3 * W * (3L)^2 = 9LW
So, the new design holds 9 times the original volume of detergent, which is 288 in.^3, so the new design holds 2592 in.^3 of detergent.