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# help geo

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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?

Jun 6, 2022

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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

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

Jun 6, 2022