Question

question 18
the picture above is of an ice-cream cone. the cone is filled with ice-cream and there is a scoop of ice-cream on the top. what is the volume of the ice-cream?
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Explanation:

Step1: Find the radius of the cone and hemisphere

The diameter of the base of the cone (and the hemisphere) is 12 in, so the radius \( r=\frac{12}{2} = 6 \) in.

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 \( h = 17 \) in and \( r = 6 \) in.
\[

$$\begin{align*} V_{cone}&=\frac{1}{3}\pi(6)^{2}(17)\\ &=\frac{1}{3}\pi\times36\times17\\ & = 12\pi\times17\\ &=204\pi \end{align*}$$

\]

Step3: Calculate the volume of the hemisphere

The formula for the volume of a sphere is \( V_{sphere}=\frac{4}{3}\pi r^{3} \), so the volume of a hemisphere is \( V_{hemisphere}=\frac{1}{2}\times\frac{4}{3}\pi r^{3}=\frac{2}{3}\pi r^{3} \).
Substituting \( r = 6 \) in:
\[

$$\begin{align*} V_{hemisphere}&=\frac{2}{3}\pi(6)^{3}\\ &=\frac{2}{3}\pi\times216\\ &= 144\pi \end{align*}$$

\]

Step4: Calculate the total volume of the ice - cream

The total volume \( V = V_{cone}+V_{hemisphere} \)
\[

$$\begin{align*} V&=204\pi + 144\pi\\ &=348\pi\\ &\approx348\times3.14\\ & = 1092.72 \end{align*}$$

\]

Answer:

\( 348\pi\) (or approximately \( 1092.72 \))