Asked by tj
A rectangular sheet of metal has identical squares cut from each corner. The sheet is then bent along the dotted lines to form an open box. The volume of the box is 420 in.3.
The equation 4x3 – 72x2 + 320x = 420 can be used to find x, the side length of the square cut from each corner.
What is the side length of the square that is cut from each corner, to the nearest inch?
All Answers 1
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GPT-5 mini
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Divide by 4: x^3 − 18x^2 + 80x − 105 = 0.
Test x = 3: it is a root, so factor to (x − 3)(x^2 − 15x + 35) = 0.
The quadratic roots are x = (15 ± √85)/2 ≈ 12.11 and ≈ 2.89.
The 12.11 in. root is impossible because the original sheet is 20 in by 16 in (L + W = 36, LW = 320), so x must be < 8. The two feasible roots (≈2.89 and 3) round to 3 in.
Answer: 3 inches.
Test x = 3: it is a root, so factor to (x − 3)(x^2 − 15x + 35) = 0.
The quadratic roots are x = (15 ± √85)/2 ≈ 12.11 and ≈ 2.89.
The 12.11 in. root is impossible because the original sheet is 20 in by 16 in (L + W = 36, LW = 320), so x must be < 8. The two feasible roots (≈2.89 and 3) round to 3 in.
Answer: 3 inches.
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