Fill in each box to make the statements true.
1 answer:
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Answer:
A. x(x +4)(3 -x) . . . . a cubic will have 3 linear factors
B. The dimensions are x, x+4, 3 -x, or 1.694 ft, 5.694 ft, 1.306 ft
C. x = (√37 -1)/3 ≈ 1.694254 ft
D. Maximum volume: ≈ 12.597 ft³
Step-by-step explanation:
A. The first attachment shows the synthetic division of the given polynomial by x+4. The result is -x^2 +3x = x(3 -x). Hence the factored polynomial is ...
x(3-x)(x+4) . . . . . . shown as positive
Since the polynomial is a cubic, we know there will be 3 roots. For this problem, that means we expect 3 factors.
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B. We are apparently to assume that the dimensions of the tank correspond to the three factors:
- x
- 3-x
- x+4
For the maximum size tank (see part C), the corresponding measurements in feet are
- 1.694 ft
- 1.306 ft
- 5.694 ft
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C. Tank volume will be maximized when the derivative of volume with respect to x is zero. The derivative polynomial is ...
dV/dx = -3x^2 -2x +12
Solving this by the usual methods, we find the positive value of x to be ...
x = (-1 +√37)/3 ≈ 1.694254 . . . . feet
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D. Evaluating the polynomial for this value of x gives ...
volume ≈ 12.5972 . . . cubic feet
