Question

an ice-cream consists of a cone and a hemisphere. the height of the cone is 14 cm and the diameter of the ice cream is 7 cm. calculate the volume of ice-cream, assuming the cone is completely filled.

Explanation:

Step1: Find the radius of the cone and hemisphere

The diameter of the ice - cream (which is the same as the diameter of the cone's base and the hemisphere) is \(d = 7\space cm\). The radius \(r=\frac{d}{2}=\frac{7}{2}=3.5\space cm\).

Step2: Calculate the volume of the cone

The formula for the volume of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\), where \(h = 14\space cm\) (height of the cone) and \(r = 3.5\space cm\).
Substitute the values:
\(V_{cone}=\frac{1}{3}\times\pi\times(3.5)^{2}\times14\)
\(=\frac{1}{3}\times\pi\times12.25\times14\)
\(=\frac{171.5}{3}\pi\)

Step3: Calculate the volume of the hemisphere

The formula for the volume of a hemisphere is \(V_{hemisphere}=\frac{2}{3}\pi r^{3}\).
Substitute \(r = 3.5\space cm\):
\(V_{hemisphere}=\frac{2}{3}\times\pi\times(3.5)^{3}\)
\(=\frac{2}{3}\times\pi\times42.875\)
\(=\frac{85.75}{3}\pi\)

Step4: Calculate the total volume of the ice - cream

The total volume \(V = V_{cone}+V_{hemisphere}\)
\(V=\frac{171.5}{3}\pi+\frac{85.75}{3}\pi=\frac{171.5 + 85.75}{3}\pi=\frac{257.25}{3}\pi\)
\(\pi=\frac{22}{7}\), so \(V=\frac{257.25}{3}\times\frac{22}{7}\)
\(257.25\div3 = 85.75\)
\(85.75\times\frac{22}{7}=\frac{85.75\times22}{7}=\frac{1886.5}{7}=269.5\space cm^{3}\)

Answer:

The volume of the ice - cream is \(269.5\space cm^{3}\) (or if we keep it in terms of \(\pi\), \(\frac{257.25}{3}\pi=\frac{85.75}{1}\pi = 85.75\pi\space cm^{3}\approx269.5\space cm^{3}\))