To obtain the total volume of ice cream, the following steps are necessary:
Step 1: Make a sketch of the shapes stated in the question, as below:
Step 2: Recall the formulas for the volume of a cone and that of a hemisphere, as follows:
The volume of a cone is:
[tex]V_{cone}=\frac{1}{3}\times\pi\times r^2\times h[/tex]And, the volume of a hemisphere is:
[tex]V_{hemisphere}=\frac{2}{3}\times\pi\times r^3[/tex]where, in both cases:
r = radius
h = perpendicular height
Step 3: Interpret the question to find clues, as follows:
"...that a half-scoop of ice cream is the shape of a hemisphere..." means that:
[tex]V_{hemisphere}=\frac{1}{2}\times V_{cone}[/tex]Thus, we have:
[tex]\begin{gathered} V_{hemisphere}=\frac{1}{2}\times V_{cone} \\ \Rightarrow\frac{2}{3}\times\pi\times r^3=\frac{1}{2}\times\frac{1}{3}\times\pi\times r^2\times h \end{gathered}[/tex]Since, the height of the cone is given to be 12 cm, we have:
[tex]\begin{gathered} \Rightarrow\frac{2}{3}\times\pi\times r^3=\frac{1}{2}\times\frac{1}{3}\times\pi\times r^2\times h \\ \Rightarrow\frac{2}{3}\times\pi\times r^3=\frac{1}{6}\times\pi\times r^2\times h \\ \text{Divide both sides by: }\frac{2}{3}\times\pi\times r^2 \\ \text{Thus:} \\ \Rightarrow\frac{\frac{2}{3}\times\pi\times r^3}{\text{ }\frac{2}{3}\times\pi\times r^2}=\frac{\frac{1}{6}\times\pi\times r^2\times h}{\text{ }\frac{2}{3}\times\pi\times r^2} \\ \Rightarrow r=\frac{1}{4}\times h \\ \sin ce\text{ h=12} \\ \Rightarrow r=\frac{1}{4}\times12=\frac{12}{4} \\ \Rightarrow r=3\operatorname{cm} \end{gathered}[/tex]Step 4: Now that we have used the clue to find the radius of the hemisphere (and cone), we can now proceed to find the total volume of the figure, as follows:
[tex]\text{Total volume of ice cream = }V_{hemisphere}+V_{cone}[/tex]Thus:
[tex]\begin{gathered} \text{Total volume of ice cream = }V_{hemisphere}+V_{cone} \\ \Rightarrow\text{Total volume of ice cream = }_{}\frac{2}{3}\times\pi\times r^3+\frac{1}{3}\times\pi\times r^2\times h \\ \Rightarrow\text{Total volume of ice cream = }_{}\frac{1}{3}\times\pi\times r^2(2r+h) \end{gathered}[/tex]Since:
h = 12 cm
r = 3cm
and pie = 3.14 (as given), we have that:
[tex]\begin{gathered} \Rightarrow\text{Total volume of ice cream = }_{}\frac{1}{3}\times\pi\times r^2(2r+h) \\ \Rightarrow\text{Total volume of ice cream = }_{}\frac{1}{3}\times3.14\times(3)^2(2(3)+12) \\ \Rightarrow\text{Total volume of ice cream = }_{}\frac{1}{3}\times3.14\times9(6+12) \\ \Rightarrow\text{Total volume of ice cream = }_{}\frac{1}{3}\times3.14\times9\times^{}(18) \\ \Rightarrow\text{Total volume of ice cream = }_{}\frac{508.68}{3}=169.56 \\ \Rightarrow\text{Total volume of ice cream = }_{}169.6\operatorname{cm}^3\text{ (to the nearest tenth)} \end{gathered}[/tex]Therefore, the total volume of ice cream is 169.6 cubic centimeter
