Question

water is poured into a rectangular container 10 centimeters (cm) long, 12 cm wide, and 9 cm high, until it is $\frac{1}{20}$ full. all this water is then poured into an empty cylindrical container with a circular base of radius 3 cm. the height of the water in the cylindrical container is $\frac{y}{pi}$ centimeters, where $y$ is a constant. what is the value of $y$?

Explanation:

Step1: Calculate volume of water in rectangular container

The volume of a rectangular - prism (the rectangular container) is $V = l\times w\times h$. Here, $l = 10$ cm, $w = 12$ cm, and $h = 9$ cm. The water fills $\frac{1}{20}$ of the rectangular container. So the volume of water $V_{water}=\frac{1}{20}\times10\times12\times9$ cm³.
$V_{water}=\frac{1}{20}\times1080 = 54$ cm³.

Step2: Calculate volume of water in cylindrical container

The volume of a cylinder is $V=\pi r^{2}h$. The radius of the cylindrical container $r = 3$ cm and the height of the water is $\frac{y}{\pi}$ cm. So $V_{cylinder}=\pi\times3^{2}\times\frac{y}{\pi}$.
$V_{cylinder}=9y$ cm³.

Step3: Equate the two volumes

Since the volume of water is the same in both containers, we set $V_{water}=V_{cylinder}$.
$54 = 9y$.

Step4: Solve for y

Divide both sides of the equation $54 = 9y$ by 9. We get $y = 6$.

Answer:

6