The distance between Sue and a dude ranch was 30 miles straight accross a rattlesnake-infested canyon. Instead of crossing the canyon, she hiked due north for a long time and then hiked due east for a ahorter leg of the journey to the ranch. If she averaged 4 mph and it took her 9 hours to get to the ranch, then how far did she hike in a northerly direction?

To find out how far Sue hiked in a northerly direction, we will use the given information, average speed, and total time.

Let's break down Sue's journey into two components: the northward hike and the eastward hike.

First, we can calculate the time it took Sue to complete the northward hike. We know that she averaged 4 mph, and the total time of her journey was 9 hours. Therefore, the time she spent on the northward hike can be calculated as follows:

Time = Total time - Time spent on the eastward hike
Time = 9 hours - Time spent on the eastward hike

Next, we need to determine the distance Sue covered during the time spent on the northward hike. Since she hiked due north for a long time, this distance will be in a straight line.

Distance = Speed × Time

Now, let's find the time spent on the eastward hike. For this, we can use the concept of relative velocity. Since the total distance she traveled is 30 miles, and the distance covered on the northward hike is part of this total distance, we can subtract the distance of the eastward hike from the total distance to get the distance of the northward hike.

Distance of the northward hike = Total distance - Distance of the eastward hike

Using the time and distance values we have obtained, we can calculate the distance of the northward hike:

Distance of the northward hike = Speed × Time

Finally, we can substitute the given information into the equations to find the distance Sue hiked in a northerly direction.

Distance of the northward hike = 4 mph × (9 hours - Time spent on the eastward hike)
Distance of the northward hike = 4 mph × (9 hours - (Total distance - Distance of the eastward hike) / Speed)

By substituting the total distance and the distance of the eastward hike values into the equation and solving it, we can determine how far Sue hiked in a northerly direction.