To determine the volume of water remaining in Abigail's cup, given that she sloshes out [tex]\(\frac{2}{3}\)[/tex] of the water, we will follow these steps:
1. Identify the given dimensions of the cone-shaped cup:
- Radius [tex]\( r = 2 \)[/tex] cm
- Height [tex]\( h = 6 \)[/tex] cm
2. Calculate the initial volume of water in the cone-shaped cup:
The formula for the volume of a cone is:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
Substituting the given dimensions:
[tex]\[
V = \frac{1}{3} \pi (2^2) (6)
\][/tex]
[tex]\[
V = \frac{1}{3} \pi \cdot 4 \cdot 6
\][/tex]
[tex]\[
V = \frac{1}{3} \pi \cdot 24
\][/tex]
[tex]\[
V = 8 \pi
\][/tex]
3. Determine the actual initial volume:
Since [tex]\( \pi \approx 3.1416 \)[/tex],
[tex]\[
V \approx 8 \cdot 3.1416 = 25.1327 \text{ cm}^3
\][/tex]
4. Calculate the volume sloshed out by Abigail:
Abigail sloshes out [tex]\(\frac{2}{3}\)[/tex] of the water from the cup. Therefore,
the volume sloshed out is:
[tex]\[
\text{Volume sloshed out} = \frac{2}{3} \times 25.1327 \approx 16.7551 \text{ cm}^3
\][/tex]
5. Find the remaining volume of water:
Subtract the volume sloshed out from the initial volume:
[tex]\[
\text{Volume remaining} = 25.1327 \text{ cm}^3 - 16.7551 \text{ cm}^3 \approx 8.3776 \text{ cm}^3
\][/tex]
Therefore, the volume of water remaining in Abigail's cup after she sloshes out [tex]\(\frac{2}{3}\)[/tex] of it is approximately [tex]\( 8.3776 \)[/tex] cm³.
