A storage unit in the shape of a rectangular prism has a volume of 72 ft3. The area of the base of the unit is 18 ft2. What is the volume of a similar unit whose height is 8 ft?
the height of the given prism = 72/18 = 4 ft
volumes of similar solids are proportional to the cube of their corresponding sides, so
x/72 = 8^3/4^3
x = 72(512)/64 = 576
or
8/4 = 2 , so the sides are twice as big
so the volume is 8 times as big (cubes of sides)
72*8 = 576
To find the volume of a similar unit with a height of 8 ft, we need to understand the relationship between the volumes of similar shapes.
The volume of a rectangular prism is given by the formula V = lwh, where l represents the length, w represents the width, and h represents the height of the prism.
In this case, the volume of the original storage unit is given as 72 ft³. Let's assume the original unit has a length of L, a width of W, and a height of H.
We know that the volume of the original unit is 72 ft³, so we can write the equation as:
72 = L * W * H ----(1)
We are also given that the area of the base of the unit is 18 ft². The area of the base of a rectangular prism is given by the formula A = lw, where A represents the area, l represents the length, and w represents the width.
Using the given information, we can write the equation as:
18 = L * W ----(2)
To find the volume of a similar unit with a height of 8 ft, we need to find the values of l and w that satisfy both equation (1) and equation (2) and then substitute h = 8 ft into the formula for volume V = lwh.
To solve these equations simultaneously, we can divide equation (1) by equation (2):
(72 / 18) = (L * W * H) / (L * W)
Simplifying this equation gives:
4 = H
This means that the height of the original unit is 4 ft.
Now we can substitute the values of L = 6 ft, W = 3 ft, and H = 4 ft into the volume formula V = lwh to find the volume of the similar unit with a height of 8 ft:
V = 6 ft * 3 ft * 8 ft
V = 144 ft³
Therefore, the volume of a similar unit with a height of 8 ft is 144 ft³.