+73 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?
+23252 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
