SOLUTION: Here's my problem... A rectangular box is 4cm longer and 3cm narrower than a certain cube. The rectangular box and the cube have equal heights and equal surface areas. Find th

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Question 274404: Here's my problem...
A rectangular box is 4cm longer and 3cm narrower than a certain cube. The rectangular box and the cube have equal heights and equal surface areas. Find the length and width of the retangular box.
I am honestly at a loss on this one.
Thanks for your help!
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
Let's start with the cube.
Let x = the side (length, width and height) of the cube. The faces of the cube are all squares. The area of one of these squares is . There are 6 faces on a cube so the total surface area is

The box has the same height as the cube so the height is x. The box's length is 4 more than the length of the cube: x+4. The width of the box is 3 less than the cube's width: x-3. The surface area of a rectangular box is 2lw + 2lh + 2wh. With our length, width and height the surface area of our box is 2(x+4)(x-3) + 2(x+4)x + 2(x-3)x.

We are told that the surface areas of the cube and box are the same so:

To solve this we start by simplifying:

Now we want the variable on just one side so we'll subtract from each side:

Add 24 to each side:

Divide both sides by 3:

Since x = side of the cube (and the height of the box), this is not our answer. We are asked to find the width, x-3, and length, x+4. So the width is 5 and the length is 12.