Question

a cone - shaped paper water cup has a height of 12 cm and a radius of 6 cm. if the cup is filled with water to five - sixths its height, what portion of the volume of the cup is filled with water? exactly □ of the cup is filled with water (type a simplified fraction)

Explanation:

Step1: Find the ratio of heights

The height of the water \(h=\frac{5}{6}\times12 = 10\) cm. Let the height of the cone \(H = 12\) cm. The ratio of heights \(r=\frac{h}{H}=\frac{5}{6}\).

Step2: Use the formula for the volume of a cone

The volume of a cone \(V=\frac{1}{3}\pi R^{2}H\). For similar cones (the water - filled part and the whole cone), if the ratio of heights is \(r\), and the ratio of radii is also \(r\) (by similar - triangles property in a cone). The volume of the water - filled cone \(V_{1}=\frac{1}{3}\pi r^{2}R^{2}\times rH\). Then \(\frac{V_{1}}{V}=r^{3}\).

Step3: Calculate the ratio of volumes

Since \(r = \frac{5}{6}\), then \(\frac{V_{1}}{V}=(\frac{5}{6})^{3}=\frac{125}{216}\)

Answer:

\(\frac{125}{216}\)