Question

this question has two parts. first, answer part a. then, answer part b.
part a
packaging a juice manufacturer is creating new cylindrical packaging. the height of the cylinder is to be 3 inches longer than the radius of the can. the cylinder is to have a volume of 628 cubic inches. use 3.14 for $\pi$.
a. write a polynomial equation to support the model.
$\pi x^3 + \square \pi x^2 = \square$
part b
b. write and solve a system of equations.
$y = \pi x^3 + \square \pi x^2$, $y = \square$; $x = \square$
c. what are the radius and height of the new packaging?
radius = $\square$ in., height = $\square$ in.

Explanation:

Step1: Define variables for radius/height

Let $x$ = radius (in inches). Then height $h = x + 3$ (in inches).

Step2: Use cylinder volume formula

Volume $V = \pi r^2 h$. Substitute $r=x$, $h=x+3$, $V=628$:
$\pi x^2(x+3) = 628$
Expand left side: $\pi x^3 + 3\pi x^2 = 628$

Step3: Set up system for Part B

The system uses the volume equation and the constant volume:
$y = \pi x^3 + 3\pi x^2$, $y = 628$

Step4: Solve for radius $x$

Substitute $\pi=3.14$ into the equation:
$3.14x^3 + 3(3.14)x^2 = 628$
Simplify: $3.14x^3 + 9.42x^2 = 628$
Divide all terms by 3.14:
$x^3 + 3x^2 = 200$
Test positive real root: $x=5$: $5^3 + 3(5)^2 = 125 + 75 = 200$, which is correct.

Step5: Calculate height

Height $h = x + 3 = 5 + 3 = 8$

Answer:

Part A

$\pi x^3 + \boldsymbol{3}\pi x^2 = \boldsymbol{628}$

Part B

$y = \pi x^3 + \boldsymbol{3}\pi x^2$, $y = \boldsymbol{628}$; $x = \boldsymbol{5}$
radius = $\boldsymbol{5}$ in., height = $\boldsymbol{8}$ in.