a rectangular prism with a volume of 400 cubic centimeters has the dimensions x + 1 centimeters, 2x…

Explanation:

Step1: Set up the system of equations

Let $y = 2x^{3}+14x^{2}+12x - 400$ and $y = 0$. Use a graphing - calculator to graph the function $y = 2x^{3}+14x^{2}+12x - 400$.

Step2: Find the x - intercepts

The x - intercepts of the graph of $y = 2x^{3}+14x^{2}+12x - 400$ are the solutions of the equation $2x^{3}+14x^{2}+12x - 400=0$. By using a graphing calculator, we find that the positive real - valued root of the equation is $x = 4$.

Step3: Calculate the side lengths

The side lengths of the rectangular prism are $x + 1$, $2x$, and $x + 6$. When $x = 4$, $x + 1=4 + 1=5$, $2x=2\times4 = 8$, and $x + 6=4+6 = 10$.

Answer:

10 centimeters