A cylindrical vase has a diameter of 6 inches. At the bottom of the vase, there are 9 marbles, each of 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?
[tex]$\pi(3 i n)^2(12 i n)-1.5(\frac{4}{3} \pi(9 i n)^3)$[/tex]
[tex]$\pi(3 i n)^2(12 i n)-9(\frac{4}{3} \pi(1.5 i n)^3)$[/tex]
[tex]$\pi(12 i n)^2(3 i n)-1.5(\frac{4}{3} \pi(9 i n)^3)$[/tex]
[tex]$\pi(12 i n)^2(3 i n)-9(\frac{4}{3} \pi(1.5 i n)^3)$[/tex]
Answer (2)
Calculate the volume of the cylinder using the formula V cy l in d er = π r 2 h , where r = 3 inches and h = 12 inches.
Calculate the volume of a single marble using the formula V s p h ere = 3 4 π r 3 , where r = 1.5 inches.
Calculate the total volume of the 9 marbles by multiplying the volume of a single marble by 9.
Subtract the total volume of the marbles from the volume of the cylinder to find the volume of the water: π ( 3 in ) 2 ( 12 in ) − 9 ( 3 4 π ( 1.5 in ) 3 ) .
Explanation
Problem Analysis Let's analyze the problem. We have a cylindrical vase with some marbles at the bottom, and the vase is filled with water. We need to find the expression that calculates the volume of water in the vase.
Calculate the volume of the cylinder First, we need to calculate the volume of the cylinder. The formula for the volume of a cylinder is given by: V cy l in d er = π r 2 h where r is the radius and h is the height. The diameter of the vase is 6 inches, so the radius is r = 2 6 = 3 inches. The height of the water is given as h = 12 inches. Therefore, the volume of the cylinder is: V cy l in d er = π ( 3 ) 2 ( 12 ) = π ( 9 ) ( 12 ) = 108 π cubic inches
Calculate the volume of a single marble Next, we need to calculate the volume of a single marble. The formula for the volume of a sphere is given by: V s p h ere = 3 4 π r 3 where r is the radius. The diameter of each marble is 3 inches, so the radius is r = 2 3 = 1.5 inches. Therefore, the volume of a single marble is: V s p h ere = 3 4 π ( 1.5 ) 3 = 3 4 π ( 3.375 ) = 4.5 π cubic inches
Calculate the total volume of the marbles Now, we need to calculate the total volume of the 9 marbles. Since each marble has a volume of 4.5 π cubic inches, the total volume of the 9 marbles is: V ma r b l es = 9 × V s p h ere = 9 × 4.5 π = 40.5 π cubic inches
Calculate the volume of the water Finally, we need to subtract the total volume of the marbles from the volume of the cylinder to find the volume of the water: V w a t er = V cy l in d er − V ma r b l es = 108 π − 40.5 π = 67.5 π cubic inches So, the expression to calculate the volume of water is: V w a t er = π ( 3 ) 2 ( 12 ) − 9 ( 3 4 π ( 1.5 ) 3 )
Find the correct option Comparing the derived expression with the given options, we can see that the correct option is: π ( 3 in ) 2 ( 12 in ) − 9 ( 3 4 π ( 1.5 in ) 3 )
Final Answer Therefore, the correct expression to calculate the volume of water in the vase is: π ( 3 in ) 2 ( 12 in ) − 9 ( 3 4 π ( 1.5 in ) 3 )
Examples
Understanding volume displacement is crucial in many real-world applications. For example, when designing ships, engineers need to calculate how much water the ship will displace to ensure it floats properly. Similarly, in medicine, understanding fluid displacement helps doctors measure lung capacity or the volume of organs. Even in everyday life, knowing how much water an object displaces can help you estimate the volume of irregularly shaped objects, like rocks for your aquarium!
The total charge delivered by a device with a current of 15.0 A for 30 seco n d s is calculated to be 450 C . This charge corresponds to approximately 2.81 × 1 0 21 electrons, as each electron has a charge of 1.6 × 1 0 − 19 C . Thus, around 2.81 × 1 0 21 electrons flow through the device.
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