Question

Recall the equation that modeled the volume of the raised flower bed, y, in terms of the width of the box, y = x3 + 11x2 − 312x. Now, open the graphing tool and graph the equation. Remember, this equation represents the volume of a flower box, so neither the width nor the volume can be negative. Using the pointer, determine the x-intercept where the width is positive and the volume will change to positive as x increases.

Answer 1

From the attached graph, the x-intercept is (13, 0)

Explanation:

The question equation is y = x³ + 11·x² - 312·x

To graph the equation, we find the value of y for a given value of x and we plot the values on a graph as follows;

x, y

13, 0

14, 532

15, 1170

16, 1920

17, 2788

18, 3780

19, 4902

20, 6160

21, 7560

22, 9108

23, 10810

24, 12672

25, 14700

From the graph, the x-intercept can be found at at x = 13.

The x-intercept can also be obtained by equating the right hand side of the equation to 0 (y = 0 is the x -intercept) as follows;

0 = x³ + 11·x² - 312·x

Therefore, x = 0 is one of the x-intercepts

By dividing by x, we have

x² + 11·x - 312 = 0

By completing the square we have

x² + 11·x + (11/2)² = 312+(11/2)²

(x + 11/2)² = 1369/4

x + 11/2 = ±37/2

x = 37/2 - 11/2 = 13 or x = -37/2 - 11/2 = -24

Which gives the x-intercept as (13, 0).