A box has a total surface area of 100. The length of the box is equal to twice its width, as well as equal to 8 less than its height. What is the height of the box?
Let the width of the box==W
The length of the box ==2W
The height ==2W + 8
Surface Area=2LW+2LH+2HW
100==2[2W*W] + 2[2W*(2W + 8)] + 2[ (2W + 8)*W]
Solve for W:
100 = 4 W^2 + 6 W (2 W + 8)
100 = 4 W^2 + 6 W (2 W + 8) is equivalent to 4 W^2 + 6 W (2 W + 8) = 100:
4 W^2 + 6 W (2 W + 8) = 100
Expand out terms of the left hand side:
16 W^2 + 48 W = 100
Divide both sides by 16:
W^2 + 3 W = 25/4
Add 9/4 to both sides:
W^2 + 3 W + 9/4 = 17/2
Write the left hand side as a square:
(W + 3/2)^2 = 17/2
Take the square root of both sides:
W + 3/2 = sqrt(17/2) or W + 3/2 = -sqrt(17/2)
Subtract 3/2 from both sides:
W = sqrt(17/2) - 3/2 or W + 3/2 = -sqrt(17/2)
Subtract 3/2 from both sides:
W = sqrt(17/2) - 3/2 ==1.415476
L ==1.415476 x 2 == 2.830952
H==2.830952 + 8 == 10.830952
Check: 2[2.830952*1.415476] + 2[2.830952*10.830952] + 2[10.830952*1.415476] ==~100
