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 288in^ 3 of detergent, how much does the new design hold? (1 point)
To find the volume of the pyramid-shaped box, we need to calculate the volume of a pyramid.
The volume of a pyramid is given by the formula V = (1/3)bh, where b is the area of the base and h is the height of the pyramid.
In this case, the base of the original rectangular box is a rectangle with dimensions length = l, width = w, and height = h. The volume of the original box is given as V = lwh = 288 in^3.
The new design uses the same amount of material for the sides but arranges it into a pyramid shape. Since the height and width of the box remain the same, we only need to find the new length of the base of the pyramid.
Using the formula for the volume of a pyramid, the new volume V' = (1/3)bh', where h' is the height of the pyramid (which is the same as the original box) and b' is the area of the new base of the pyramid.
Since the base of the pyramid is a square (since it is a pyramid inscribed in a rectangular box), the area of the base is given by b' = s^2, where s is the length of the side of the square base.
Since the volume of the pyramid is the same as the original box, we have (1/3)s^2h = 288 in^3.
Rearranging for s^2, we get s^2 = 3 * 288/h.
Since the base is a square, the length of each side is s = √(3 * 288/h).
Substitute this back into the volume formula for the pyramid, V' = (1/3)(√(3 * 288/h))^2 * h = 288/3 = 96 in^3.
Therefore, the new design of the pyramid-shaped box holds 96in^3 of detergent.