Question
Consider this container of popcorn. Do not round the numbers.
(3 points)
Part 1. Volume of the container.
Determine the radius of the container.
inches
Determine the area of the circular base of the container.
square inches (Remember to use 3.14 for pi.)
Determine the volume of the container.
cubic inches
Question 2
Suppose you want to place 6 containers in a rectangular prism that is about 15 inches tall, 22 inches wide, and 7.6 inches wide.(2 points)
Part 2: Volume of the Rectangular Prism.
Determine the area of the base of the rectangular prism.
square inches
Determine the volume of the rectangular prism.
cubic inches
Question 3
Part 3. Will the 6 popcorn containers fit? If so, how much space is left?(1 point)
Responses
No, the containers need about 250 more square inches of space.
No, the containers need about 250 more square inches of space.
No, the containers need about 1,500 more square inches.
No, the containers need about 1,500 more square inches.
Yes, there will be about 1,008 square inches of space left.
Yes, there will be about 1,008 square inches of space left.
Yes, there will be about 2,258 square inches of space left.
(3 points)
Part 1. Volume of the container.
Determine the radius of the container.
inches
Determine the area of the circular base of the container.
square inches (Remember to use 3.14 for pi.)
Determine the volume of the container.
cubic inches
Question 2
Suppose you want to place 6 containers in a rectangular prism that is about 15 inches tall, 22 inches wide, and 7.6 inches wide.(2 points)
Part 2: Volume of the Rectangular Prism.
Determine the area of the base of the rectangular prism.
square inches
Determine the volume of the rectangular prism.
cubic inches
Question 3
Part 3. Will the 6 popcorn containers fit? If so, how much space is left?(1 point)
Responses
No, the containers need about 250 more square inches of space.
No, the containers need about 250 more square inches of space.
No, the containers need about 1,500 more square inches.
No, the containers need about 1,500 more square inches.
Yes, there will be about 1,008 square inches of space left.
Yes, there will be about 1,008 square inches of space left.
Yes, there will be about 2,258 square inches of space left.
Answers
GPT-4o mini
To solve these parts, we need to calculate the volume of the popcorn container, the volume of the rectangular prism, and then determine if the containers will fit inside the prism.
### Part 1: Volume of the Container
1. **Determine the radius of the container.**
- Let's assume we are given the radius, for the sake of explanation, that the radius \( r \) is, for example, 3 inches.
2. **Determine the area of the circular base of the container.**
\[
\text{Area} = \pi r^2
\]
Substituting \( \pi \) as 3.14 and \( r = 3 \):
\[
\text{Area} = 3.14 \times (3^2) = 3.14 \times 9 = 28.26 \text{ square inches}
\]
3. **Determine the volume of the container.**
- If the height \( h \) is, for example, 12 inches:
\[
\text{Volume} = \text{Area of base} \times \text{height} = 28.26 \times 12 = 339.12 \text{ cubic inches}
\]
### Part 2: Volume of the Rectangular Prism
1. **Determine the area of the base of the rectangular prism.**
- The dimensions of the rectangular prism are 15 inches tall, 22 inches wide, and 7.6 inches deep. The base area is calculated using the width and depth.
\[
\text{Area of base} = \text{width} \times \text{depth} = 22 \times 7.6 = 167.2 \text{ square inches}
\]
2. **Determine the volume of the rectangular prism.**
\[
\text{Volume} = \text{Area of base} \times \text{height} = 167.2 \times 15 = 2508 \text{ cubic inches}
\]
### Part 3: Will the 6 popcorn containers fit?
- First, calculate the total volume of the 6 popcorn containers. If the volume of each container is 339.12 cubic inches:
\[
\text{Total volume of 6 containers} = 6 \times 339.12 = 2034.72 \text{ cubic inches}
\]
- Now, we can see if they fit in the prism:
- **Volume of the rectangular prism:** 2508 cubic inches
- **Volume of the containers:** 2034.72 cubic inches
- To find out how much space is left:
\[
\text{Space left} = \text{Volume of rectangular prism} - \text{Total volume of containers} = 2508 - 2034.72 = 473.28 \text{ cubic inches}
\]
### Conclusion
If the volume of the 6 containers is less than the volume of the prism, they will fit.
Based on the calculations:
- **Yes, the containers will fit.**
- There will be about **473.28 cubic inches left** inside the prism.
Since to answer your question options provided regarding space, the closest option could not match with our calculation, as we have approximately 473.28 cubic inches remaining which isn't present in the provided choices.
If you have specific values for the radius and height of the container, please share them for more accurate calculations!
### Part 1: Volume of the Container
1. **Determine the radius of the container.**
- Let's assume we are given the radius, for the sake of explanation, that the radius \( r \) is, for example, 3 inches.
2. **Determine the area of the circular base of the container.**
\[
\text{Area} = \pi r^2
\]
Substituting \( \pi \) as 3.14 and \( r = 3 \):
\[
\text{Area} = 3.14 \times (3^2) = 3.14 \times 9 = 28.26 \text{ square inches}
\]
3. **Determine the volume of the container.**
- If the height \( h \) is, for example, 12 inches:
\[
\text{Volume} = \text{Area of base} \times \text{height} = 28.26 \times 12 = 339.12 \text{ cubic inches}
\]
### Part 2: Volume of the Rectangular Prism
1. **Determine the area of the base of the rectangular prism.**
- The dimensions of the rectangular prism are 15 inches tall, 22 inches wide, and 7.6 inches deep. The base area is calculated using the width and depth.
\[
\text{Area of base} = \text{width} \times \text{depth} = 22 \times 7.6 = 167.2 \text{ square inches}
\]
2. **Determine the volume of the rectangular prism.**
\[
\text{Volume} = \text{Area of base} \times \text{height} = 167.2 \times 15 = 2508 \text{ cubic inches}
\]
### Part 3: Will the 6 popcorn containers fit?
- First, calculate the total volume of the 6 popcorn containers. If the volume of each container is 339.12 cubic inches:
\[
\text{Total volume of 6 containers} = 6 \times 339.12 = 2034.72 \text{ cubic inches}
\]
- Now, we can see if they fit in the prism:
- **Volume of the rectangular prism:** 2508 cubic inches
- **Volume of the containers:** 2034.72 cubic inches
- To find out how much space is left:
\[
\text{Space left} = \text{Volume of rectangular prism} - \text{Total volume of containers} = 2508 - 2034.72 = 473.28 \text{ cubic inches}
\]
### Conclusion
If the volume of the 6 containers is less than the volume of the prism, they will fit.
Based on the calculations:
- **Yes, the containers will fit.**
- There will be about **473.28 cubic inches left** inside the prism.
Since to answer your question options provided regarding space, the closest option could not match with our calculation, as we have approximately 473.28 cubic inches remaining which isn't present in the provided choices.
If you have specific values for the radius and height of the container, please share them for more accurate calculations!