To find the rate of the second bus, we start with the given equation:
\[ 3(r + 10) + 3r = 282 \]
First, we can distribute the \(3\) on the left side:
\[ 3r + 30 + 3r = 282 \]
Now, combine the terms with \(r\):
\[ 6r + 30 = 282 \]
Next, we isolate \(6r\) by subtracting \(30\) from both sides:
\[ 6r = 282 - 30 \]
\[ 6r = 252 \]
Now, we divide both sides by \(6\) to solve for \(r\):
\[ r = \frac{252}{6} \]
\[ r = 42 \]
Thus, the rate of the second bus is \( \boxed{42} \) miles per hour.
To verify, we can calculate the speed of the first bus and the distance traveled by both buses in 3 hours. The first bus travels at:
\[ r + 10 = 42 + 10 = 52 \text{ miles per hour} \]
In 3 hours, the first bus travels:
\[ 3 \times 52 = 156 \text{ miles} \]
The second bus travels:
\[ 3 \times 42 = 126 \text{ miles} \]
The total distance apart after 3 hours is:
\[ 156 + 126 = 282 \text{ miles} \]
The calculations confirm that the distances add up correctly, therefore the rate of the second bus is indeed \( \boxed{42} \) miles per hour.
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