Asked by Christine
The dimensions of a gift box are consecutive positive integers such that the height is the least integer and the length is the greatest integer. If the height is increased by 1 cm, the width is increased by 2 cm, and the length is increased by 3 cm, then a larger box is constructed such that the volume is increased by 456 cm^3. Determine the dimensions of each box.
Answers
Answered by
Reiny
Original box
height: x
width : x+1
length: x+2
Volume = x(x+1)(x+2)
new box
height:x+1
width: x+3
length: x+5
volume = (x+1)(x+3)(x+5)
so (x+1)(x+3)(x+5) - x(x+1)(x+2) = 456
which when expanded, then simplified gave me the quadratic
2x^2 + 7x - 147 = 0
which factors to
(x-7)(2x+21) = 0
so x=7 or x = -21/2
but x had to be an integer, so x=7
sub back in to get the dimensions
height: x
width : x+1
length: x+2
Volume = x(x+1)(x+2)
new box
height:x+1
width: x+3
length: x+5
volume = (x+1)(x+3)(x+5)
so (x+1)(x+3)(x+5) - x(x+1)(x+2) = 456
which when expanded, then simplified gave me the quadratic
2x^2 + 7x - 147 = 0
which factors to
(x-7)(2x+21) = 0
so x=7 or x = -21/2
but x had to be an integer, so x=7
sub back in to get the dimensions
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.