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Question 355745: An open box with a square base is to have a volume of 12ft^3
Find a function that models the surface area of the box.
Found 2 solutions by Alan3354, Theo:
Answer by Alan3354(69443)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! The volume of a rectangular prism is L*W*H which can be the same formula to model the volume of a box which is a rectangular prism.
If the dimensions of the base are L*W, and the base is square, this means that L = W.
We'll make L equal to S and W equal to S since they are both equal to each other.
The volume of the box becomes S*S*H which simplifies to S^2*H.
The surface area of the box would be the sum of all the area of the sides of the box.
The area of the base is equal to S^2
The area of each side is S*H.
The surface area of the box would then be equal to 2*S^2 + 4*H*S.
Since the top of the box is open, then we need to subtract it's area.
This makes our formula equal to 1*S^2 + 4*H*S which can be simplified further to S^2 + 4*H*S.
The formula for the Surface Area of the box would be S^2 + 4*H*S
That would be your formula.
S is equal to measurement of each side of the base.
H is equal to the height.
Volume of the Box = S^2*H.
If the volume of the box is 12 cubic feet, then:
12 = S^2*H
Surface Area = S^2 + 4*H*S
We can solve for one of the variables in the volume formula.
We'll solve for H.
Since 12 = S^2*H, then H = 12/S^2
If we substitute 12/S^2 for H in the surface area formula, then we get:
S^2 + 4*H*S = Surface Area of the Box = S^2 + (4*12*S)/S^2)
This simplifies to:
Surface area of the box = S^2 + (48/S)
Since S^2 is equivalent to S^3/S, then this equation can be made into:
Surface area of the box = (S^3/S) + (48/S) which can be further simplified into:
Surface area of the box = (S^3 + 48)/S.
This assumes that the volume of the box is equal to 12.
To see if this accurate, let's assume that each side of the base of the box is equal to 2 feet and the height of the box is equal to 3 feet.
The volume of the box would be S^2*H = 4*3 = 12.
That part is good.
The surface area of the box would be equal to (S^3 + 48)/S.
Since S is equal to 2, this becomes (8+48)/2 which becomes 56/2 which becomes 28.
The numbers check out so the formula is good.
The area of the base is equal to 2*2 = 4.
The area of each side is equal to 2*3 = 6
There is one base and 4 sides to be equal to 4 + 24 = 28.
The area of the top of the box is missing as instructed by the problem statement.
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