Answer:
[tex]\sf C. \quad \pi (3\:in)^2(12\:in)-9 \left(\dfrac{4}{3} \pi (1.5\:in)^3 \right)[/tex]
Step-by-step explanation:
To find the volume of water in the vase, subtract the volume of the marbles from the volume of the cylindrical vase.
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The marbles can be modeled as spheres.
Volume of a sphere
[tex]\sf V=\dfrac{4}{3}\pi r^3 \quad \textsf{(where r is the radius)}[/tex]
Radius
[tex]\sf r = \dfrac{1}{2}d \quad \textsf{(where d is the diameter)}[/tex]
Therefore, the volume of 9 marbles, each of diameter 3 inches is:
[tex]\sf Volume & = \sf 9 \left(\dfrac{4}{3} \pi (1.5\:in)^3 \right)[/tex]
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The vase can be modeled as a cylinder.
Volume of a cylinder
[tex]\sf V=\pi r^2 h \quad \textsf{(where r is the radius and h is the height)}[/tex]
Therefore, the volume of a cylinder with a base of diameter 6 in and a height of 12 in is:
[tex]\sf Volume =\pi (3\:in)^2(12\:in)[/tex]
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Volume of water
[tex]\begin{aligned}\implies \sf Volume\:of\:water & = \sf Volume\:of\:vase- Volume\:of\:marbles\\& = \sf \pi (3\:in)^2(12\:in)-9 \left(\dfrac{4}{3} \pi (1.5\:in)^3 \right)\end{aligned}[/tex]
