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avatar+190 

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 5 less than its height. If the total surface area of the box is 234, then what is the height of the box?

 Apr 3, 2024
 #1
avatar+128794 
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L = H  - 5

H = L + 5

 

L = 2W

W = L/2

W = ( H - 5) / 2

 

SA = 2 [ LW + HL + HW ]

 

234  = 2  [  (H - 5)(H - 5)/2 + H(H  -5)  +  H (H - 5)  / 2 ]

 

234  = (H - 5)(H-5)  + 2H (H -5)  + H(H -5)

 

234 =  H^2 - 10H + 25 + 2H^2 - 10H + H^2 - 5H

 

4H^2  - 25H  + 25  = 234

 

4H^2  - 25H - 209 =  0

 

H  =   [25 + sqrt [ 625 -  4*4 * -209 ] ] / 8   =  [25 + sqrt [ 3969 ] / 8  = [25 + 63 ] / 8  = 88 / 8  =  11

 

cool cool cool

 Apr 4, 2024
 #2
avatar+2 
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To find the height of the box, use the given relations in the problem and the formula for the total surface area of a rectangular box, work through algebraic simplifications and potentially systems of equations to solve for the dimensions, which would give the height.

 Apr 4, 2024

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