Question

question 5 of 5 type the correct answer in the box. use numerals instead of words. a company wants to explore packaging for its product. the company plans to place a hemisphere on top of a 6 - inch wide cylindrical can. if the volume of the package is 108π cubic inches, what is the height of the can? the height of the can is inches.

Explanation:

Step1: Find the radius of the cylinder

The diameter of the cylindrical part is 6 inches, so the radius $r=\frac{6}{2}=3$ inches.

Step2: Write the volume formula for the combined - shape

The volume of the combined shape (cylinder + hemisphere) is $V = V_{cylinder}+V_{hemisphere}$. The volume of a cylinder is $V_{cylinder}=\pi r^{2}h$ and the volume of a hemisphere is $V_{hemisphere}=\frac{2}{3}\pi r^{3}$. So $V=\pi r^{2}h+\frac{2}{3}\pi r^{3}$.

Step3: Substitute the known values

We know that $V = 108\pi$ and $r = 3$. Substitute these values into the formula:
\[

$$\begin{align*} 108\pi&=\pi\times3^{2}\times h+\frac{2}{3}\pi\times3^{3}\\ 108\pi&=9\pi h + 18\pi \end{align*}$$

\]

Step4: Solve for h

First, divide both sides of the equation by $\pi$: $108=9h + 18$. Then subtract 18 from both sides: $9h=108 - 18=90$. Finally, divide both sides by 9: $h = 10$.

Answer:

10