A snow cone is a tasty treat with flavored ice and a spherical bubble gum ball at the bottom, as shown below:
The radius of the cone is 1.5 inches, and its height is 3 inches. If the diameter of the bubble gum ball is 0.5 inches, what is the closest approximation of the volume of the cone that can be filled with flan?
[tex]21.21 in ^3[/tex]
[tex]0.07 in ^3[/tex]
[tex]7.00 in ^3[/tex]
[tex]7.07 in ^3[/tex]
Answer (1)
Calculate the volume of the cone using the formula V co n e = 3 1 π r 2 h , where r = 1.5 inches and h = 3 inches.
Calculate the volume of the spherical bubble gum ball using the formula V s p h ere = 3 4 π r 3 , where the radius of the sphere is r = 2 0.5 = 0.25 inches.
Subtract the volume of the sphere from the volume of the cone to find the volume of the cone that can be filled with flan: V f l an = V co n e − V s p h ere .
Approximate the value of V f l an to the nearest hundredth. 7.00 i n 3
Explanation
Problem Analysis We are given a snow cone consisting of a cone filled with flavored ice and a spherical bubble gum ball at the bottom. We need to find the volume of the cone that can be filled with flan, which is the volume of the cone minus the volume of the bubble gum ball.
Calculate the volume of the cone First, let's calculate the volume of the cone. The formula for the volume of a cone is given by: V co n e = 3 1 π r 2 h where r is the radius of the cone and h is the height of the cone. We are given that the radius of the cone is r = 1.5 inches and the height is h = 3 inches. Plugging these values into the formula, we get: V co n e = 3 1 π ( 1.5 ) 2 ( 3 ) = 3 1 π ( 2.25 ) ( 3 ) = π ( 2.25 ) = 2.25 π
Calculate the volume of the sphere Next, let's calculate the volume of the spherical bubble gum ball. 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 of the sphere. We are given that the diameter of the bubble gum ball is 0.5 inches, so the radius is r = 2 0.5 = 0.25 inches. Plugging this value into the formula, we get: V s p h ere = 3 4 π ( 0.25 ) 3 = 3 4 π ( 0.015625 ) = 3 4 ( 0.015625 ) π = 3 0.0625 π
Subtract the volumes Now, we need to subtract the volume of the sphere from the volume of the cone to find the volume of the cone that can be filled with flan: V f l an = V co n e − V s p h ere = 2.25 π − 3 0.0625 π = ( 2.25 − 3 0.0625 ) π V f l an = ( 2.25 − 0.0208333... ) π = 2.2291666... π Using the value of π ≈ 3.14159 , we have: V f l an = 2.2291666... × 3.14159 ≈ 7.003133623627247
Approximate the final volume Finally, we need to approximate the value of V f l an to the nearest hundredth. We have V f l an ≈ 7.003133623627247 , so rounding to the nearest hundredth, we get V f l an ≈ 7.00 cubic inches.
Final Answer The closest approximation of the volume of the cone that can be filled with flan is 7.00 cubic inches.
Examples
Snow cones are a popular treat, and understanding their volume helps vendors determine how much flavored ice and other ingredients they need. For example, knowing the volume allows them to accurately measure the amount of syrup needed to flavor the ice, ensuring each snow cone tastes delicious and is cost-effective to produce. This also helps in pricing the snow cones appropriately based on the ingredients used.
