Question

A cylindrical base has a diameter of 6 inches. At the bottom of the vase, there are 9 marbles, each of the diameter 3 inches. The vase is filled with water up to a height of 12 inches. Which of the following could be used to calculate the volume of water in the vase please help

Answer 1

Answer:

$\sf C. \quad \pi (3\:in)^2(12\:in)-9 \left(\dfrac{4}{3} \pi (1.5\:in)^3 \right)$

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

$\sf V=\dfrac{4}{3}\pi r^3 \quad \textsf{(where r is the radius)}$

Radius

$\sf r = \dfrac{1}{2}d \quad \textsf{(where d is the diameter)}$

Therefore, the volume of 9 marbles, each of diameter 3 inches is:

$\sf Volume & = \sf 9 \left(\dfrac{4}{3} \pi (1.5\:in)^3 \right)$

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The vase can be modeled as a cylinder.

Volume of a cylinder

$\sf V=\pi r^2 h \quad \textsf{(where r is the radius and h is the height)}$

Therefore, the volume of a cylinder with a base of diameter 6 in and a height of 12 in is:

$\sf Volume =\pi (3\:in)^2(12\:in)$

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Volume of water

$\begin{aligned}\implies \sf Volume\:of\:water & = \sf Volume\:of\:vase- Volume\:of\:marbles\\& = \sf \pi (3\:in)^2(12\:in)-9 \left(\dfrac{4}{3} \pi (1.5\:in)^3 \right)\end{aligned}$