To solve the problem of calculating the volume of water in the vase, we start by analyzing it step by step.
1. Vase Specifications:
- The vase is cylindrical.
- Diameter of the vase: 4 inches.
- Height of the vase: 8 inches.
2. Volume of the Vase:
- The radius (r) of the vase is half of the diameter, so [tex]\( r = \frac{4}{2} = 2 \)[/tex] inches.
- Volume of a cylinder is given by the formula [tex]\( V = \pi r^2 h \)[/tex].
- Substituting the given values, we get the volume of the vase as:
[tex]\[
V_{\text{vase}} = \pi (2 \, \text{in})^2 (8 \, \text{in}) = \pi \cdot 4 \, \text{in}^2 \cdot 8 \, \text{in} = 32\pi \, \text{in}^3
\][/tex]
3. Marbles Specifications:
- Each marble has a diameter of 3 inches.
- Radius of each marble [tex]\( r_{\text{marble}} = \frac{3}{2} = 1.5 \)[/tex] inches.
- There are 6 marbles.
4. Volume of a Single Marble:
- The volume of a sphere is given by [tex]\( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)[/tex].
- Substituting the radius of each marble, we get:
[tex]\[
V_{\text{marble}} = \frac{4}{3} \pi (1.5 \, \text{in})^3
\][/tex]
5. Volume of 6 Marbles:
- Since there are 6 marbles, we multiply the volume of one marble by 6:
[tex]\[
V_{\text{6 marbles}} = 6 \left( \frac{4}{3} \pi (1.5 \, \text{in})^3 \right)
\][/tex]
6. Volume of Water in the Vase:
- The total volume of water in the vase is the volume of the vase minus the volume occupied by the 6 marbles:
[tex]\[
V_{\text{water}} = V_{\text{vase}} - V_{\text{6 marbles}} = 32\pi \, \text{in}^3 - 6 \left( \frac{4}{3} \pi (1.5 \, \text{in})^3 \right)
\][/tex]
Given the choices, the correct expression for the volume of water in the vase is:
[tex]\[
\pi (2 \, \text{in})^2 (8 \, \text{in}) - 6 \left( \frac{4}{3} \pi (1.5 \, \text{in})^3 \right)
\][/tex]
This matches the first option:
[tex]\[
\pi(2 \, \text{in})^2(8 in)-6\left(\frac{4}{3} \pi (1.5 in)^3 \right)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\pi(2 \, \text{in})^2(8 in)-6\left(\frac{4}{3} \pi (1.5 in)^3 \right)}
\][/tex]
