Question 230565
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: {{{V = l*w*h}}}. Since your volume is 10000 and the length and width are x and the height are y:
{{{10000 = x*x*y}}}
or
{{{10000 = x^2y}}}
We can solve this for y:
{{{10000/x^2 = y}}}<br>
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,<ol><li>Draw a box with square ends</li><li>Label the sides of the square "x"</li><li>Label the edges of the box which connect the squares to each other "y".</li></ol>
Area of a square: {{{A = s^2}}}
The area of your square, since the sides are x, is {{{x^2}}}
Area of a rectangle: {{{A = l*w}}}
The area of your rectangles, since the length and width are x and y: {{{xy}}}.
The total surface area is the sum of the areas of the 2 squares and the 4 rectangles:
{{{SA = 2(x^2) + 4(xy) = 2x^2 + 4xy}}}
Since we want this in terms of x, we need to substitute for the "y". Earlier we found, from the volume, that {{{y = 10000/x^2}}}. Substituting this for y in our surface area equation we get:
{{{SA = 2x^2 + 4x(10000/x^2)}}}
which simplifies to:
{{{SA = 2x^2 + 40000/x}}}