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The inside of an ice cream cone is filled with cream and has height 12 cm Assuming that a half-scoop of ice cream is the shape of a hemisphere and that it fits perfectly on top of the cone (same radius) find the total volume of ice cream. Use 3.14 for pie and round your answer to the nearest tenth.

User Melchia
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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:


V_(cone)=(1)/(3)*\pi* r^2* h

And, the volume of a hemisphere is:


V_(hemisphere)=(2)/(3)*\pi* r^3

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:


V_(hemisphere)=(1)/(2)* V_(cone)

Thus, we have:


\begin{gathered} V_(hemisphere)=(1)/(2)* V_(cone) \\ \Rightarrow(2)/(3)*\pi* r^3=(1)/(2)*(1)/(3)*\pi* r^2* h \end{gathered}

Since, the height of the cone is given to be 12 cm, we have:


\begin{gathered} \Rightarrow(2)/(3)*\pi* r^3=(1)/(2)*(1)/(3)*\pi* r^2* h \\ \Rightarrow(2)/(3)*\pi* r^3=(1)/(6)*\pi* r^2* h \\ \text{Divide both sides by: }(2)/(3)*\pi* r^2 \\ \text{Thus:} \\ \Rightarrow\frac{(2)/(3)*\pi* r^3}{\text{ }(2)/(3)*\pi* r^2}=\frac{(1)/(6)*\pi* r^2* h}{\text{ }(2)/(3)*\pi* r^2} \\ \Rightarrow r=(1)/(4)* h \\ \sin ce\text{ h=12} \\ \Rightarrow r=(1)/(4)*12=(12)/(4) \\ \Rightarrow r=3\operatorname{cm} \end{gathered}

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:


\text{Total volume of ice cream = }V_(hemisphere)+V_(cone)

Thus:


\begin{gathered} \text{Total volume of ice cream = }V_(hemisphere)+V_(cone) \\ \Rightarrow\text{Total volume of ice cream = }_{}(2)/(3)*\pi* r^3+(1)/(3)*\pi* r^2* h \\ \Rightarrow\text{Total volume of ice cream = }_{}(1)/(3)*\pi* r^2(2r+h) \end{gathered}

Since:

h = 12 cm

r = 3cm

and pie = 3.14 (as given), we have that:


\begin{gathered} \Rightarrow\text{Total volume of ice cream = }_{}(1)/(3)*\pi* r^2(2r+h) \\ \Rightarrow\text{Total volume of ice cream = }_{}(1)/(3)*3.14*(3)^2(2(3)+12) \\ \Rightarrow\text{Total volume of ice cream = }_{}(1)/(3)*3.14*9(6+12) \\ \Rightarrow\text{Total volume of ice cream = }_{}(1)/(3)*3.14*9*^{}(18) \\ \Rightarrow\text{Total volume of ice cream = }_{}(508.68)/(3)=169.56 \\ \Rightarrow\text{Total volume of ice cream = }_{}169.6\operatorname{cm}^3\text{ (to the nearest tenth)} \end{gathered}

Therefore, the total volume of ice cream is 169.6 cubic centimeter

The inside of an ice cream cone is filled with cream and has height 12 cm Assuming-example-1
User Wanda
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