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

Sammy can use the slope formula to show that the opposite sides have equal slopes.

Step-by-step explanation:

If EFGH is a parallelogram then its opposite sides should be parallel and equal

We know that slopes of parallel lines are equal .

So, we need to find the slopes of parallel lines so, that we can show that their slopes are equal

So, Option C is true

Sammy can use the slope formula to show that the opposite sides have equal slopes.

So, she can use coordinate geometry to determine whether EFGH is a parallelogram by using slope formula to show that the opposite sides have equal slopes.