Question 1209114
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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 = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-6+-sqrt((6)^2-4(1)(-120)))/(2(1))}}}


{{{x = (-6+-sqrt(36 + 480))/(2)}}}


{{{x = (-6+-sqrt(516))/(2)}}}


{{{x = (-6+-sqrt(4*129))/(2)}}}


{{{x = (-6+-sqrt(4)*sqrt(129))/(2)}}}


{{{x = (-6+- 2*sqrt(129))/(2)}}}


{{{x = (2(-3+-sqrt(129)))/(2)}}}


{{{x = -3+-sqrt(129)}}}


{{{x = -3+sqrt(129)}}} or {{{x = -3-sqrt(129)}}}


{{{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+8 = -3+sqrt(129)+8 = 5+sqrt(129) = 16.357816691601}}}
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