A box has length 3x cm, width 2x cm and height x cm. The total surface area of the box is 2662 cm^2. Find an equation for the surface area of the box, and use this to find the true values of the dimensions of the box.

Here we are given the surface are of a cuboid box and the dimensions of the box with an unknown variable. Well the total surface area of a cuboid, is going to be the surface area of each of the faces of it added together.

We know there are 6 faces, so we should get 6 terms involving x adding up to make the surface area. The first faces, will be the faces at the side of the cuboid. These are going to have the same surface area, equal to the area of the rectangles that they're made up of. So 2HeightWidth = 22xx = 4x^2. The next two faces are on the top and bottom of the cuboid, these are the lengthwidth, and there are two of these rectangles: 23x2x = 12x^2. Lastly, the front and back facing rectangles form the rest of the cuboid's surface area. These are the lengthheight of the shape: 23xx = 6x^2. Summing these up gives, 22x^2 as the total surface area. Our equationis therefore, 22x^2 = 2662. Solving this, first divide through both sides by 22. 2662/22 = 121 (using a calculator or by a hand method). Now we have that x^2 = 121. If we take the square root of both sides, we get that: x = 11 as our value of x.

To work out the actual dimensions of the box, substitute this back in to the information we were given at the start. Height= 111= 11cm, Length= 311=33cm and width= 2*11=22cm And now we're done.