For a cuboid, 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. (a) Construct an equation to calculate the surface area.

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First label all the sides with letters:

longest side = a

middle length side = b

shortest side = c

Now convert the question into equation form:

"the longest side is two units more than the shortest side" becomes, a = 2 + c (1)

"middle length side is one unit longer than the shortest side" becomes, b = 1 + c (2)

"The total surface area of the cuboid is 52 units² " becomes, 2*a*b + 2*a*c + 2*b*c = 52 (3)

Substitute expressions (1) and (2) into (3) to get

2(2+c)(1+c) + 2c(2+c) + 2c(1+c) = 52

Expand out the brackets:

2(2+3c+c^2) + 4c + 2c^2 + 2c + 2c^2 = 52

which becomes:

4 + 6c + 2c^2 + 6c + 4c^2 = 52

collect all the terms together:

6c^2 + 12c - 48 = 0

Divide by 6 to simplify equation:

c^2 + 2c - 8 = 0

factorise this equation into brackets to find c, by finding two numbers that multiply together to make -8, and add together to make 2... i.e. 4 and -2,

(c + 4)(c-2) = 0

Therefore, c = -4, or c=2. Since c must be positive (a negative side length would make no physical sense) c must be 2. We can then subsitute this back into (1) and (2) to find values for a and b,

a = 4

b = 3

and then double check that the surface area is 52.

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