To find out when the two machines will have filled the same number of cans of soup, we can set up equations based on the information provided.
Let \( t \) represent the number of minutes after the current point in time.
Machine 1:
- Initially filled: 92 cans
- Rate: 16 cans per minute
- Number of cans filled after \( t \) minutes: \( 92 + 16t \)
Machine 2:
- Initially filled: 104 cans
- Rate: 13 cans per minute
- Number of cans filled after \( t \) minutes: \( 104 + 13t \)
We want to find \( t \) such that the number of cans filled by both machines is the same:
\[ 92 + 16t = 104 + 13t \]
Now, let's solve the equation for \( t \):
- Subtract \( 13t \) from both sides:
\[ 92 + 16t - 13t = 104 \]
Which simplifies to:
\[ 92 + 3t = 104 \]
- Subtract 92 from both sides:
\[ 3t = 104 - 92 \]
This simplifies to:
\[ 3t = 12 \]
- Divide both sides by 3:
\[ t = \frac{12}{3} = 4 \]
Now that we have \( t = 4 \), it means that in 4 minutes, both machines will have filled the same number of cans.
To find out how many cans they will have filled at that time, we can substitute \( t = 4 \) back into either of the equations:
Using Machine 1:
\[ 92 + 16(4) = 92 + 64 = 156 \]
Using Machine 2:
\[ 104 + 13(4) = 104 + 52 = 156 \]
Thus, after 4 minutes, both machines will have filled 156 cans of soup.
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