Question
Sergio has two similar cylindrical pots, Pot A and Pot B. The radius of Pot A is 15 inches, and the radius of Pot B is 3 inches. What is the ratio of the volume of Pot A to the volume of Pot B?
15:3
45:9
125:1
5:1
15:3
45:9
125:1
5:1
Answers
Answered by
GPT-4o mini
To find the ratio of the volumes of the two cylindrical pots, we can use the formula for the volume \( V \) of a cylinder, which is given by:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder. Since both pots are similar, we can assume they have the same height (we'll denote it as \( h \)). This means the height cancels out when we find the ratio of the volumes.
Let's denote the radius of Pot A as \( r_A = 15 \) inches and the radius of Pot B as \( r_B = 3 \) inches.
Now we can find the volumes of both pots:
\[
V_A = \pi r_A^2 h = \pi (15)^2 h = \pi (225) h
\]
\[
V_B = \pi r_B^2 h = \pi (3)^2 h = \pi (9) h
\]
Next, we find the ratio of the volumes:
\[
\text{Ratio} = \frac{V_A}{V_B} = \frac{\pi (225) h}{\pi (9) h}
\]
The \( \pi \) and \( h \) terms cancel out:
\[
\text{Ratio} = \frac{225}{9}
\]
Now, calculating \( \frac{225}{9} \):
\[
\frac{225}{9} = 25
\]
Thus, the ratio of the volumes of Pot A to Pot B is:
\[
25:1
\]
Looking at the options provided:
- 15:3
- 45:9
- 125:1
- 5:1
None of the options match \( 25:1 \). However, since we are looking for the simplest form, \( 25:1 \) can be derived from looking at the volumes and confirming our calculations.
If we had also simplified \( 225:9 \), we would also derive that \( 25:1 \) is equivalent.
Conclusively, if looking for your listed options as a form of multiple-choice selection, please check if there is an error; otherwise, the correct volume ratio calculation finding shows \( 25:1 \), thus you could confirm or inquire about the listed choices again.
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder. Since both pots are similar, we can assume they have the same height (we'll denote it as \( h \)). This means the height cancels out when we find the ratio of the volumes.
Let's denote the radius of Pot A as \( r_A = 15 \) inches and the radius of Pot B as \( r_B = 3 \) inches.
Now we can find the volumes of both pots:
\[
V_A = \pi r_A^2 h = \pi (15)^2 h = \pi (225) h
\]
\[
V_B = \pi r_B^2 h = \pi (3)^2 h = \pi (9) h
\]
Next, we find the ratio of the volumes:
\[
\text{Ratio} = \frac{V_A}{V_B} = \frac{\pi (225) h}{\pi (9) h}
\]
The \( \pi \) and \( h \) terms cancel out:
\[
\text{Ratio} = \frac{225}{9}
\]
Now, calculating \( \frac{225}{9} \):
\[
\frac{225}{9} = 25
\]
Thus, the ratio of the volumes of Pot A to Pot B is:
\[
25:1
\]
Looking at the options provided:
- 15:3
- 45:9
- 125:1
- 5:1
None of the options match \( 25:1 \). However, since we are looking for the simplest form, \( 25:1 \) can be derived from looking at the volumes and confirming our calculations.
If we had also simplified \( 225:9 \), we would also derive that \( 25:1 \) is equivalent.
Conclusively, if looking for your listed options as a form of multiple-choice selection, please check if there is an error; otherwise, the correct volume ratio calculation finding shows \( 25:1 \), thus you could confirm or inquire about the listed choices again.