Question

The polynomial 2x^3 + 9x^2 + 4x - 15 represents the volume in cubic feet of a rectangular holding tank at a fish hatchery. The depth of the tank is (x-1) feet. The length is 13 feet.
A. Use synthetic division to help you factor the volume polynomial. How many linear factors should you look for? What are they?
B. Assume the length is the greatest dimension. Which linear factor represents the 13 ft length? What are the dimensions of the tank? What is it's volume? What is the value of x? Do you get the same volume if you substitute the volume of x into 2x^3 + 9x^2 + 4x -15?
C. Using a graphing calculator, analyze the graph. What is the relationship between the graph and the linear factors of the polynomial?

Answer 1

Answer:

A. Three factors; length, width, and depth = (2x + 5)(x + 3)(x - 1)

B. Length = 2x + 5; x = 4; Dimensions = 13 ft × 7 ft × 3 ft; V = 273 ft³

C. See below

Step-by-step explanation:

A. Synthetic division

 2   11  15      

1)2   9   4  -15

      2  11   15

  2  11  15    0

So, (2x³ + 9x² + 4x – 15)/(x – 1) = 2x² + 11x + 15

There should be three factors, corresponding to length, width, and depth of the tank.

The depth of the tank is x - 1.

The other two factors are the length and the width of the tank. They are the factors of the quadratic.  

x² + 11x + 15 = (2x+5)(x+3)

So, the linear factors of the equation are (x-1)(2x+5)(x+3).

B. Volume of tank

(i) Identify the length factor.

2x + 5 = 2 × 13 +5 = 31

 x + 3 = 13 + 2 = 16

l > w, so  l = 2x+5

(ii) Value of x

         l =  13, so

2x + 5 = 13     Subtract 5 from each side

     2x = 8       Divide each side by 2

       x = 4

(iii) Dimensions of tank

        l = 13 ft

       w = x + 3

           = 4 + 3

           = 7 ft

       d = x – 1

          = 4 – 1

          = 3 ft

The tank dimensions are 13 ft × 7 ft × 3 ft.

(iv) Volume of tank

V = lwd

  = 13 × 7 × 3

  = 273 ft³

(v) Check

Insert the value of x into the cubic equation.

V = 2x³    + 9x²     + 4x     – 15

  = 2(4)³ + 9(4)² + 4(4) – 15

  = 128    + 144    + 16     – 15

  = 273

Substituting into the cubic gives the same volume as substituting into its factors.

(C) Graph

The graph of the function is shown below.

Only values in the first quadrant are important, because the dimensions and the volume must be positive.

The y-intercept at x = 1 corresponds to the factor representing the

depth (d = x – 1).

The green dot represents the volume when x = 4.