Albert has some flour stored in a cylindrical jar. The flour completely fills the jar, which has a diameter of 4 inches and a height of 6 inches. Select the containers that could h... Albert has some flour stored in a cylindrical jar. The flour completely fills the jar, which has a diameter of 4 inches and a height of 6 inches. Select the containers that could hold all of the flour in Albert's jar. The choices are: A container shaped like a cylinder with a radius of 2 inches and a height of 6 inches, a container shaped like a square prism with side lengths of 4 inches on its base and a height of 6 inches, a container shaped like a cone with a diameter of 4 inches and a height of 6 inches, and a container shaped like a sphere with a diameter of 4 inches.
Understand the Problem
The question asks us to calculate the volume of a cylinder and then determine which of the following containers could hold the same volume. This requires calculating the volume of a cylinder, square prism, cone, and sphere and comparing them to the original cylinder.
Answer
- A container shaped like a cylinder with a radius of 2 inches and a height of 6 inches. - A container shaped like a square prism with side lengths of 4 inches on its base and a height of 6 inches.
Answer for screen readers
The containers that could hold all of the flour in Albert's jar are:
- A container shaped like a cylinder with a radius of 2 inches and a height of 6 inches.
- A container shaped like a square prism with side lengths of 4 inches on its base and a height of 6 inches.
Steps to Solve
- Calculate the volume of the cylindrical jar
The formula for the volume of a cylinder is $V = \pi r^2 h$, where $r$ is the radius and $h$ is the height. Given the diameter is 4 inches, the radius is 2 inches. The height is 6 inches. Therefore, the volume of the jar is: $$V = \pi (2^2)(6) = 24\pi \text{ cubic inches}$$
- Calculate the volume of the first cylindrical container
The first container is a cylinder with a radius of 2 inches and a height of 6 inches. Using the same formula as above, the volume is: $$V = \pi (2^2)(6) = 24\pi \text{ cubic inches}$$
- Calculate the volume of the square prism container
The second container is a square prism with side lengths of 4 inches and a height of 6 inches. The formula for the volume of a square prism is $V = s^2 h$, where $s$ is the side length and $h$ is the height. Therefore, the volume is: $$V = (4^2)(6) = 96 \text{ cubic inches}$$
- Calculate the volume of the cone container
The third container is a cone with a diameter of 4 inches (radius of 2 inches) and a height of 6 inches. The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$. Therefore, the volume is: $$V = \frac{1}{3} \pi (2^2) (6) = 8\pi \text{ cubic inches}$$
- Calculate the volume of the sphere container
The fourth container is a sphere with a diameter of 4 inches (radius of 2 inches). The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$. Therefore, the volume is: $$V = \frac{4}{3} \pi (2^3) = \frac{32}{3} \pi \text{ cubic inches}$$
- Compare the volumes
The original jar has a volume of $24\pi$ cubic inches.
The first cylindrical container has a volume of $24\pi$ cubic inches.
The square prism container has a volume of 96 cubic inches. Since $\pi \approx 3.14$, $24\pi \approx 24*3.14 \approx 75.36$. The volume of the square prism is greater than the volume of the jar.
The cone container has a volume of $8\pi$ cubic inches. Since $\pi \approx 3.14$, $8\pi \approx 8 * 3.14 \approx 25.12$. The volume of the cone is less than the volume of the jar.
The sphere container has a volume of $\frac{32}{3}\pi$ cubic inches. Since $\pi \approx 3.14$, $\frac{32}{3}\pi \approx \frac{32}{3} * 3.14 \approx 33.49$. The volume of the sphere is smaller than the volume of the jar.
- Determine which containers can hold the flour
The first cylindrical container and the square prism container can hold all the flour.
The containers that could hold all of the flour in Albert's jar are:
- A container shaped like a cylinder with a radius of 2 inches and a height of 6 inches.
- A container shaped like a square prism with side lengths of 4 inches on its base and a height of 6 inches.
More Information
The value of $\pi$ (pi) is approximately 3.14159. It is the ratio of a circle's circumference to its diameter.
Tips
A common mistake is forgetting the $\frac{1}{3}$ factor in the volume of a cone formula. Also, using the diameter instead of the radius in the formulas can lead to incorrect calculations. Another mistake is comparing the volumes without calculating them accurately, especially when $\pi$ is involved.
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