A rectangular prism has a volume of 48 cubic feet and surface area of 88 square feet. what are its dimensions?
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To find the dimensions of the rectangular prism, we can use the given volume and surface area.
Let's denote the length, width, and height of the rectangular prism as l, w, and h, respectively.
We are given that the volume of the prism is 48 cubic feet. The volume of a rectangular prism is calculated by multiplying its length, width, and height. Therefore, we have:
Volume = l * w * h = 48
We are also given that the surface area of the prism is 88 square feet. The surface area of a rectangular prism is calculated by summing the areas of all its six faces. Therefore, we have:
Surface Area = 2lw + 2lh + 2wh = 88
Now we have a system of two equations:
Equation 1: l * w * h = 48
Equation 2: 2lw + 2lh + 2wh = 88
We can use these equations to solve for the dimensions of the rectangular prism.
To make it easier, we can solve Equation 1 for h:
h = 48 / (lw)
Now we substitute this value in Equation 2:
2lw + 2l(48/(lw)) + 2w(48/(lw)) = 88
Simplifying the equation further:
2lw + 96/l + 96/w = 88
Multiplying through by lw to eliminate the fractions:
2l^2w + 96w + 96l = 88lw
Rearranging the equation:
2l^2w - 88lw + 96w + 96l = 0
Factoring the equation:
2w(l^2 - 44l + 48) + 96l = 0
Now we can solve for l by setting each factor equal to zero:
2w = 0 or (l^2 - 44l + 48) = 0
If we set 2w = 0, it means w is zero, which is not physically possible for a rectangular prism. Therefore, we can ignore this factor.
Now we can solve the quadratic equation (l^2 - 44l + 48) = 0 using factoring or the quadratic formula.
Once we find the possible values of l, we can substitute them back into Equation 1 to find corresponding values of w and h.