Question
How many glasses like the one shown can be filled three-quarters full from a pitcher holding 150 cubic inches of water? The diameter of the cup is 4 inches and the height is 4 inches
Answers
GPT-4o mini
To determine how many glasses can be filled three-quarters full from a pitcher holding 150 cubic inches of water, we first need to calculate the volume of one glass.
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
1. **Calculate the radius of the glass:**
- The diameter of the glass is given as 4 inches, so the radius \( r \) is:
\[
r = \frac{d}{2} = \frac{4}{2} = 2 \text{ inches}
\]
2. **Use the height of the glass:**
- The height \( h \) is given as 4 inches.
3. **Calculate the volume of the glass:**
- Plug the values into the formula:
\[
V = \pi (2)^2 (4) = \pi (4)(4) = 16\pi \text{ cubic inches}
\]
4. **Calculate the volume when the glass is filled three-quarters full:**
- The volume when filled three-quarters full is:
\[
\text{Volume}_{\text{3/4 full}} = \frac{3}{4} V = \frac{3}{4} (16\pi) = 12\pi \text{ cubic inches}
\]
5. **Calculate how many glasses can be filled with the pitcher:**
- First, we need to approximate the value of \( \pi \). Using \( \pi \approx 3.14 \):
\[
12\pi \approx 12 \times 3.14 = 37.68 \text{ cubic inches}
\]
6. **Calculate the number of glasses that can be filled:**
- Now, divide the total volume of the pitcher by the volume of water needed for one glass filled three-quarters full:
\[
\text{Number of glasses} = \frac{150}{12\pi} \approx \frac{150}{37.68} \approx 3.98
\]
Since we can only fill whole glasses, we round down to the nearest whole number, yielding 3 glasses.
Thus, the number of glasses that can be filled three-quarters full from a pitcher holding 150 cubic inches of water is \( \boxed{3} \).
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
1. **Calculate the radius of the glass:**
- The diameter of the glass is given as 4 inches, so the radius \( r \) is:
\[
r = \frac{d}{2} = \frac{4}{2} = 2 \text{ inches}
\]
2. **Use the height of the glass:**
- The height \( h \) is given as 4 inches.
3. **Calculate the volume of the glass:**
- Plug the values into the formula:
\[
V = \pi (2)^2 (4) = \pi (4)(4) = 16\pi \text{ cubic inches}
\]
4. **Calculate the volume when the glass is filled three-quarters full:**
- The volume when filled three-quarters full is:
\[
\text{Volume}_{\text{3/4 full}} = \frac{3}{4} V = \frac{3}{4} (16\pi) = 12\pi \text{ cubic inches}
\]
5. **Calculate how many glasses can be filled with the pitcher:**
- First, we need to approximate the value of \( \pi \). Using \( \pi \approx 3.14 \):
\[
12\pi \approx 12 \times 3.14 = 37.68 \text{ cubic inches}
\]
6. **Calculate the number of glasses that can be filled:**
- Now, divide the total volume of the pitcher by the volume of water needed for one glass filled three-quarters full:
\[
\text{Number of glasses} = \frac{150}{12\pi} \approx \frac{150}{37.68} \approx 3.98
\]
Since we can only fill whole glasses, we round down to the nearest whole number, yielding 3 glasses.
Thus, the number of glasses that can be filled three-quarters full from a pitcher holding 150 cubic inches of water is \( \boxed{3} \).