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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 3 less than its height. What is the height of the box?

 Apr 20, 2020
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length  =  L

width  =  W

height  =  H

 

length = twice the width        --->   L  =  2W   

length  = 3 less than height   --->   L  =  H - 3     

 

Since  L  =  2W  and  L  =  H - 3   (combining them)   --->   H - 3  =  2W   --->   W  =  (H - 3) / 2

 

The total surface area is 100   --->   2LW + 2LH + 2 HW  =  100   --->   LW + LH + HW  =  50

Substituting:   ( H - 3 )( (H - 3)/2) + ( H - 3 )(H) + (H)( (H - 3)/2 )  =  50   

--->    ( H2 - 6H + 9 ) / 2  +  H2 - 3H  + ( H2 - 3H ) / 2  =  50

Multiplying by 2:

--->   H2 - 6H + 9 + 2H2 - 6H + H2 - 3H  =  100

--->   4H2 - 15H - 91  =  0

--->   (H - 7)(4H + 13)  =  0

--->   H  =  7

 Apr 20, 2020

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