Question

amelia is making mini jars full of homemade chocolate syrup to give out as party favors. she makes a big batch of chocolate syrup in a cylindrical saucepan. the diameter of the saucepan is 12 inches, and the syrup fills the pan to a height of 3 inches. the jars she plans to fill are also cylindrical, with a diameter of 3 inches and a height of 2.9 inches.
what is the maximum number of jars that amelia can completely fill using the chocolate syrup?

jars

Explanation:

Step1: Calculate the volume of the chocolate syrup in the saucepan

The formula for the volume of a cylinder is \(V = \pi r^{2}h\).
The diameter of the saucepan \(d = 12\) inches, so the radius \(r=\frac{d}{2}=\frac{12}{2} = 6\) inches, and the height \(h = 3\) inches.

$$V_{saucepan}=\pi\times6^{2}\times3=\pi\times36\times 3=108\pi$$

Step2: Calculate the volume of one jar

The diameter of the jar \(d = 3\) inches, so the radius \(r=\frac{d}{2}=\frac{3}{2}=1.5\) inches, and the height \(h = 2.9\) inches.

$$V_{jar}=\pi\times(1.5)^{2}\times2.9=\pi\times2.25\times2.9 = 6.525\pi$$

Step3: Find the number of jars

Divide the volume of the syrup in the saucepan by the volume of one jar: \(n=\frac{V_{saucepan}}{V_{jar}}=\frac{108\pi}{6.525\pi}\)
Since \(\pi\) cancels out (\(\frac{108}{6.525}\approx16.55\))

Answer:

\(16\)