Two buses leave towns 369 miles apart at the same time and travel toward each other. One bus travels \( 9 \frac{\mathrm{mi}}{\mathrm{h}} \) faster than the other. If they meet in 3 hours, what is the rate of each bus? Note that the Alexs graphing calculator can be used to make computations easier. Rate of the faster bus:

To solve this, let the speed of the slower bus be \( x \) miles per hour. Then, the speed of the faster bus is \( x + 9 \) miles per hour. Since they are traveling toward each other, their combined speed is \( x + (x + 9) = 2x + 9 \) miles per hour. Given they meet after 3 hours, the total distance covered is: \[ (2x + 9) \times 3 = 369 \] Simplifying: \[ 6x + 27 = 369 \] Subtract 27 from both sides: \[ 6x = 342 \] Now, divide by 6: \[ x = 57 \] So the slower bus travels at 57 miles per hour, and the faster bus, which is \( x + 9 \), travels at \( 57 + 9 = 66 \) miles per hour. Rate of the faster bus: **66 miles per hour.** To double-check this, you can simply take the speeds of both buses, 57 and 66, and see how far they go in 3 hours. The slower bus covers \( 57 \times 3 = 171 \) miles, and the faster one covers \( 66 \times 3 = 198 \) miles. Add these two distances together: \( 171 + 198 = 369 \) miles, confirming our calculations are spot on!