Question
Riley set up a snack table for her dog's birthday party. She invited all the neighborhood dogsβand their owners! For the dogs, she filled small toy balls with peanut butter. The balls have a radius of 1.75 inches. What is the volume of each ball?Use βπβ3.14 and round your answer to the nearest whole number.cubic inchesFor her neighbors, Riley filled mini cone-shaped bags with popcorn. Riley notices that the bags have the same radius as the balls, but a volume of 44 cubic inches. What is the height of each bag?Use βπβ3.14 and round your answer to the nearest whole number.inches
Answers
Answered by
GPT-4o mini
To find the volume of each ball, we can use the formula for the volume of a sphere, which is given by:
\[
V = \frac{4}{3} \pi r^3
\]
where \(r\) is the radius of the sphere.
Given that the radius of the balls is \(1.75\) inches, we calculate the volume as follows:
1. Calculate \(r^3\):
\[
r^3 = (1.75)^3 = 1.75 \times 1.75 \times 1.75 = 5.359375
\]
2. Plug \(r^3\) into the volume formula:
\[
V = \frac{4}{3} \pi (1.75)^3 \approx \frac{4}{3} \times 3.14 \times 5.359375
\]
3. Calculate \(\frac{4}{3} \times 3.14\):
\[
\frac{4}{3} \times 3.14 \approx 4.18667
\]
4. Finally, calculate the volume:
\[
V \approx 4.18667 \times 5.359375 \approx 22.425
\]
Rounding this to the nearest whole number gives us:
\[
\text{Volume of each ball} \approx 22 \text{ cubic inches}
\]
Next, to find the height of each mini cone-shaped bag, we can use the formula for the volume of a cone:
\[
V = \frac{1}{3} \pi r^2 h
\]
Given that the volume \(V\) is \(44\) cubic inches and the radius \(r\) is \(1.75\) inches, we can rearrange the equation to solve for \(h\):
1. Calculate \(r^2\):
\[
r^2 = (1.75)^2 = 3.0625
\]
2. Plug values into the volume formula:
\[
44 = \frac{1}{3} \pi (1.75^2) h \implies 44 = \frac{1}{3} \times 3.14 \times 3.0625 \times h
\]
3. Simplify \(\frac{1}{3} \times 3.14 \times 3.0625\):
\[
\frac{1}{3} \times 3.14 \approx 1.04667
\]
\[
1.04667 \times 3.0625 \approx 3.200141875
\]
So we have:
\[
44 = 3.200141875 h
\]
4. Solve for \(h\):
\[
h = \frac{44}{3.200141875} \approx 13.75
\]
Rounding this to the nearest whole number gives us:
\[
\text{Height of each bag} \approx 14 \text{ inches}
\]
In summary:
- The volume of each ball is approximately **22 cubic inches**.
- The height of each bag is approximately **14 inches**.
\[
V = \frac{4}{3} \pi r^3
\]
where \(r\) is the radius of the sphere.
Given that the radius of the balls is \(1.75\) inches, we calculate the volume as follows:
1. Calculate \(r^3\):
\[
r^3 = (1.75)^3 = 1.75 \times 1.75 \times 1.75 = 5.359375
\]
2. Plug \(r^3\) into the volume formula:
\[
V = \frac{4}{3} \pi (1.75)^3 \approx \frac{4}{3} \times 3.14 \times 5.359375
\]
3. Calculate \(\frac{4}{3} \times 3.14\):
\[
\frac{4}{3} \times 3.14 \approx 4.18667
\]
4. Finally, calculate the volume:
\[
V \approx 4.18667 \times 5.359375 \approx 22.425
\]
Rounding this to the nearest whole number gives us:
\[
\text{Volume of each ball} \approx 22 \text{ cubic inches}
\]
Next, to find the height of each mini cone-shaped bag, we can use the formula for the volume of a cone:
\[
V = \frac{1}{3} \pi r^2 h
\]
Given that the volume \(V\) is \(44\) cubic inches and the radius \(r\) is \(1.75\) inches, we can rearrange the equation to solve for \(h\):
1. Calculate \(r^2\):
\[
r^2 = (1.75)^2 = 3.0625
\]
2. Plug values into the volume formula:
\[
44 = \frac{1}{3} \pi (1.75^2) h \implies 44 = \frac{1}{3} \times 3.14 \times 3.0625 \times h
\]
3. Simplify \(\frac{1}{3} \times 3.14 \times 3.0625\):
\[
\frac{1}{3} \times 3.14 \approx 1.04667
\]
\[
1.04667 \times 3.0625 \approx 3.200141875
\]
So we have:
\[
44 = 3.200141875 h
\]
4. Solve for \(h\):
\[
h = \frac{44}{3.200141875} \approx 13.75
\]
Rounding this to the nearest whole number gives us:
\[
\text{Height of each bag} \approx 14 \text{ inches}
\]
In summary:
- The volume of each ball is approximately **22 cubic inches**.
- The height of each bag is approximately **14 inches**.
There are no human answers yet. A form for humans to post answers is coming very soon!