Irving lives in Appletown, and plans to drive along Highway 42, a straight highway that leads to Bananatown, located 133 miles east and 23 miles north. Carol lives in Coconutville, located 84 miles east and 42 miles south of Appletown. Highway 86 runs directly north from Coconutville, and junctions with Highway 42 before heading further north to Durianville. Carol and Irving are planning to meet up at park-and-ride at the junction of the highways and carpool to Bananatown. Irving leaves Appletown at $8 \mathrm{am}$, driving his usual 45 miles per hour. If Carol leaves leaves Coconutville at 9 am, how fast will she need to drive to arrive at the park-

Step 1 ：First, we need to calculate the distance Irving needs to travel to reach the junction. This can be done using the Pythagorean theorem, as we know the eastward and northward distances. The eastward distance is 133 miles and the northward distance is 23 miles. Using the Pythagorean theorem, we get \(\sqrt{133^2 + 23^2} = 134.97\) miles.

Step 2 ：Next, we need to calculate the time it will take for Irving to reach the junction. This can be done by dividing the distance by his speed. Irving's speed is 45 miles per hour. So, the time it takes for Irving to reach the junction is \(\frac{134.97}{45} = 2.99\) hours.

Step 3 ：Finally, we need to calculate the speed Carol needs to drive to reach the junction at the same time as Irving. This can be done by dividing the distance she needs to travel by the time it takes for Irving to reach the junction. The distance Carol needs to travel is 126 miles. So, the speed Carol needs to drive is \(\frac{126}{2.99} = 42.01\) miles per hour.

Step 4 ：Final Answer: Carol needs to drive approximately \(\boxed{42.01}\) miles per hour to arrive at the park-and-ride at the same time as Irving.