Question

alberto is designing a container in the shape of a rectangular prism to ship electronic devices. the length of the container is 10 inches longer than the height. the sum of the length, width, and height is 25 inches. write a function for the volume of the prism. what do the x - intercepts of the graph mean in this context? what dimensions of the container will maximize the volume? write a function for the volume of the prism. f(x)=-2x³ - 5x²+150x (simplify your answer.) what are the x - intercepts? (simplify your answer. use a comma to separate answers as needed.)

Explanation:

Step1: Find x - intercepts

Set $f(x)=0$, so $- 2x^{3}-5x^{2}+150x = 0$. Factor out $-x$: $-x(2x^{2}+5x - 150)=0$.

Step2: Solve for x

We have two cases. First, $-x = 0$ gives $x = 0$. For the quadratic equation $2x^{2}+5x - 150=0$, use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ where $a = 2$, $b = 5$, and $c=-150$.
$x=\frac{-5\pm\sqrt{5^{2}-4\times2\times(-150)}}{2\times2}=\frac{-5\pm\sqrt{25 + 1200}}{4}=\frac{-5\pm\sqrt{1225}}{4}=\frac{-5\pm35}{4}$.
We get $x=\frac{-5 + 35}{4}=\frac{30}{4}=7.5$ and $x=\frac{-5 - 35}{4}=\frac{-40}{4}=-10$.

Answer:

$0,7.5,-10$