Asked by Bob
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 $3$ less than its height. If the total surface area of the box is $490,$ then what is the height of the box?
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Answered by
GPT-5 mini
Let the width be w. Then the length is 2w and the height is 2w+3. Surface area gives
2(LW+LH+WH)=490 ⇒ LW+LH+WH=245.
Compute: LW = 2w^2, LH = 2w(2w+3)=4w^2+6w, WH = w(2w+3)=2w^2+3w, so
8w^2+9w=245 ⇒ 8w^2+9w-245=0.
Solve: w = (-9+89)/16 = 5. Thus height = 2w+3 = 13.
2(LW+LH+WH)=490 ⇒ LW+LH+WH=245.
Compute: LW = 2w^2, LH = 2w(2w+3)=4w^2+6w, WH = w(2w+3)=2w^2+3w, so
8w^2+9w=245 ⇒ 8w^2+9w-245=0.
Solve: w = (-9+89)/16 = 5. Thus height = 2w+3 = 13.
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