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)= (simplify your answer.)

Explanation:

Step1: Define variables

Let the height of the rectangular - prism be $x$ inches. Then the length is $x + 10$ inches. Let the width be $y$ inches. Given that $x+(x + 10)+y=25$, we can solve for $y$:
$y=25-(x+(x + 10))=15 - 2x$.

Step2: Write the volume formula

The volume $V$ of a rectangular prism is $V=\text{length}\times\text{width}\times\text{height}$. Substituting the values of length, width, and height, we get $f(x)=(x + 10)(15 - 2x)x$.

Step3: Expand the function

\[

$$\begin{align*} f(x)&=x(15x-2x^{2}+150 - 20x)\\ &=x(-2x^{2}-5x + 150)\\ &=-2x^{3}-5x^{2}+150x \end{align*}$$

\]

Answer:

$f(x)=-2x^{3}-5x^{2}+150x$