A cuboid has sides such that the longest side is two units more than the shortest side, and the middle length side is one unit longer than the shortest side. The total surface area of the cuboid is 52 units². What is the length of the shortest side?

A cuboid has sides such that the longest side is two units more than the shortest side, and the middle length side is one unit longer than the shortest side. The total surface area of the cuboid is 52 units². What is the length of the shortest side We want to start by trying to visualise this cuboid. The question is made tricker by not giving us a diagram, but we can draw our own! Draw a standard cuboid on the whiteboard and label the sides: shortest side, medium side, longest side. Let's simplify this by saying our shortest side is just 'x'. The information in the question tells us about the lengths of the other two sides: they are x+1 and x+2 What else does the question tell us? The total surface area is 52. So what does surface area mean? In a cuboid this will work out as 6 areas added together. This is actually just 3 pairs of sides, so we need to work out 3 surface areas and double it. How do we work out the areas? x * (x+1) x * (x+2) (x+1) * (x+2) We can expand these brackets out: x * (x+1) = x^2 + x x * (x+2) = x^2 + 2x (x+1) * (x+2) = x^2 + 3x + 2 We need to add all of these together: x^2 + x + x^2 + 2x + x^2 + 3x + 2 = 3(x^2) + 6x +2 Remember we said we needed to double it: 6(x^2) + 2x + 4 And we know that this is equal to 52. This is from a non-calculator paper so we have to decide what method we want to use to solve this quadratic equation. We can look at the 'factorising' method. Re-write our equation so it is equal to 0: 6(x^2) + 12x - 48 = 0 We can simplify this by dividing everything by 6: (x^2) + 2x - 8 = 0 The factorising method means we need two numbers that add up to get 2, and multiple together to get -8 In this case these answers are (-2,4) So we can re-write the equation as: (x+4)(x-2)=0 Which means: x = 2 or x = -4 Because we're talking about a real cuboid, we can't have a length of -4. So our length x must be 2 units long.