Answer:
a. length: 20-2x; width: 12-2x; height: x
b. V = 4x^3 -64x^2 +240x
c. cubic trinomial
d. 262.7 cubic inches maximum for a 2.4 inch square
e. 15 × 7 × 2.5 inches for 262.5 in³ volume
Explanation:
a.
If the square cut from each corner has side length x, the dimensions are ...
- height: x
- width: 12 -2x
- length: 20 -2x
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b.
The volume is the product of length, width, and height.
V = LWH
V = (20 -2x)(12-2x)(x) = 4(x-10)(x-6)x = 4(x^2 -16x +60)x
V = 4x^3 -64x^2 +240x
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c.
The volume function is a cubic trinomial.
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d.
The maximum volume will be found where the derivative of the volume function is zero.
V' = 12x^2 -128x +240 = 0
x^2 -32/3x +20 = 0 . . . . divide by 12
(x -16/3)^2 = 76/9 . . . . . complete the square
x = (16 -√76)/3 ≈ 2.4 . . . . size of the square
V = 4(2.4)(10 -2.4)(6 -2.4) ≈ 262.7 . . . cubic inches
The maximum box volume is about 262.7 cubic inches when the square is about 2.4 inches on each side.
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e.
A graphing calculator shows there to be two solutions for a volume of 262.5 cubic inches. The rational solution is x = 2.5 inches. The box dimensions would be ...
- length = 20 -2(2.5) = 15 inches
- width = 12 -2(2.5) = 7 inches
- height = 2.5 inches