SOLUTION: Express the surface area S of a closed box with a volume of 10000 cubic inches as a function of x. The answer he has is S=2x^2+40000/x. I need to know how he reached this conclusio

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The only way your teacher's answer is correct is if the box has two faces which are squares measuring x on each side. So
Let x = the length and the width of the box
Let y = the height of the box.
Volume of a box (aka right rectangular prism) is length times width times height: . Since your volume is 10000 and the length and width are x and the height are y:

or

We can solve this for y:


Moving on to the surface area. Any closed box has 6 faces which have area. In your box you have 2 squares and 4 rectangles. The sides of the squares are x. The length and width of the rectangles are x and y. If you have trouble understanding this,

1. Draw a box with square ends
2. Label the sides of the square "x"
3. Label the edges of the box which connect the squares to each other "y".

Area of a square:
The area of your square, since the sides are x, is
Area of a rectangle:
The area of your rectangles, since the length and width are x and y: .
The total surface area is the sum of the areas of the 2 squares and the 4 rectangles:

Since we want this in terms of x, we need to substitute for the "y". Earlier we found, from the volume, that . Substituting this for y in our surface area equation we get:

which simplifies to: