SOLUTION: A box has a certain length, width, and height. The length of the box is equal to twice its width, as well as equal to 8 less than its height. If the total surface area of the box

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: A box has a certain length, width, and height. The length of the box is equal to twice its width, as well as equal to 8 less than its height. If the total surface area of the box      Log On


   

Question 1209114: A box has a certain length, width, and height. The length of the box is equal to twice its width, as well as equal to 8 less than its height. If the total surface area of the box is 480, then what is the height of the box?
Answer by math_tutor2020(3838) About Me  (Show Source):
You can put this solution on YOUR website!

Answer:
exact height = 5 + sqrt(129)
approximate height = 16.357816691601


Work Shown

"length is twice the width" ---> "width is half the length".
"length is 8 units smaller than the height" ---> "height is 8 units longer compared to length".
x = length
x/2 = 0.5x = width
x+8 = height

S = surface area of the box
S = 2*(length*width + length*height + width*height)
480 = 2*(x*0.5x + x*(x+8) + 0.5x*(x+8))
480 = 2*(0.5x^2 + x^2+8x + 0.5x^2+4x)
480 = 2*(2x^2 +12x)
480 = 4x^2 +24x
4x^2+24x-480 = 0
4(x^2+6x-120) = 0
x^2+6x-120 = 0/4
x^2+6x-120 = 0

Plug a = 1, b = 6, c = -120 into the quadratic formula.
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-6%2B-sqrt%28%286%29%5E2-4%281%29%28-120%29%29%29%2F%282%281%29%29

x+=+%28-6%2B-sqrt%2836+%2B+480%29%29%2F%282%29

x+=+%28-6%2B-sqrt%28516%29%29%2F%282%29

x+=+%28-6%2B-sqrt%284%2A129%29%29%2F%282%29

x+=+%28-6%2B-sqrt%284%29%2Asqrt%28129%29%29%2F%282%29

x+=+%28-6%2B-+2%2Asqrt%28129%29%29%2F%282%29

x+=+%282%28-3%2B-sqrt%28129%29%29%29%2F%282%29

x+=+-3%2B-sqrt%28129%29

x+=+-3%2Bsqrt%28129%29 or x+=+-3-sqrt%28129%29

x+=+8.357816691601 or x+=+-14.357816691601
Ignore the negative x value.
A negative length makes no sense.
Each decimal value mentioned is approximate.
Round it however your teacher instructs.

Add 8 to the length to find the height.
height+=+x%2B8+=+-3%2Bsqrt%28129%29%2B8+=+5%2Bsqrt%28129%29+=+16.357816691601