The polynomial -x^3-x^2+12x represents the volume of a rectangular aquatic tank in cubic feet. the length of the tank is (x+4). a. use synthetic formula to factor the volume of the polynomial. how many linear factors should you look for? b. what are the dimensions of the tank? c. find the value of x that will maximize the volume of the box. d. what is the maximum volume?

  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:

For the maximum size tank (see part C), the corresponding measurements in feet are

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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