College Algebra & Trigonometry, 2018a

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4.9. OPTIMIZATION 243 Geometry/Cost Optimization A closed box with a square base is built from material that costs $1 per ft 2 for the four sides and the bottom and $5 per ft 2 for the top. What are the dimensions of the box of largest volume that can be constructed for $72? There are two issues we are working with in this problem - the volume of the box and the surface area of the box. The volume of the box is important because that’s what we’re looking to maximize. The surface area of the box is important because that’s what will control the cost of the box - notice that the costs are given in terms of ft 2 or square feet which is related to the surface area. The box has a square base, but the height is some other dimension - h. So the volume of the box will be length ∗ width ∗ height, but since the box has a square base the length and width will be the same - we’ll call them x. From this we see that the volume can be expressed as: V = x ∗ x ∗ h = x 2 h For the cost of the box, we need to know the surface area. Each side of the box should be included - the base and the top have the same area (x 2 ) because they are both squares that are x units on each side. The four sides of the box are all rectangles that are x by h units. That means each one has an area of x ∗ h -soall four would be 4 ∗ x ∗ h or 4xh. This makes the surface area of the box: The cost of the box then will be: S =2x 2 +4xh C = x 2 ($1) + x 2 ($5) + 4xh($1) Here we see the one x 2 is multiplied by $1, because the bottom will cost $1 per ft 2 , but the other x 2 is multiplied by $5, because the top costs $5 per ft 2 . The four sides : 4xh is also multiplied by $1.
244 CHAPTER 4. FUNCTIONS So, our final formula for the cost is: C =6x 2 +4xh but we already know that we want to spend $72 on the box, so we can say that: 72 = 6x 2 +4xh This means that : 72 − 6x 2 4x or = h 18 x − 1.5x = h The reason why this is important is that it will allow us to express the volume of the box in terms of x. Remember that the volume was: V = x 2 h so now we can see that: ( ) 18 V = x 2 x − 1.5x or V =18x − 1.5x 3
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College Algebra & Trigonometry
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c○2018 Richard W. Beveridge This
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College Algebra & Trigonometry, 2018a

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