A woman drove her car on a road running due north at a given rate for 6 hours. She continued her trip on an unsaved road due east for 5 hours at a rate of 4mph slower. Had she been able to go to her destination in a straight line from her starting point and at her original rate she would have taken 71/2 hours. Find her rate for the first part of the journey.
(6 x)^2 + [5 (x - 4)]^2 = (7.5 x)^2
To solve this problem, we'll start by setting up a system of equations based on the given information.
Let's assume that the woman's rate for the first part of the journey (on the road running north) is r mph. Since she drove for 6 hours at this rate, the distance she covered on this road is 6r miles.
For the second part of the journey (on the unsaved road running east), she drove for 5 hours at a rate of 4 mph slower than her original rate. Therefore, her rate for the second part of the journey is (r - 4) mph. The distance covered on this road is (5 * (r - 4)) miles.
Now, according to the problem, if she had been able to go to her destination in a straight line from her starting point, she would have taken 7.5 hours. This means the total distance covered in the straight line would be the same as the total distance covered on the two separate roads.
So, we can set up the following equation:
6r + 5(r - 4) = 7.5r
Now, let's simplify and solve for r:
6r + 5r - 20 = 7.5r
11r - 20 = 7.5r
11r - 7.5r = 20
3.5r = 20
r = 20 / 3.5
r ≈ 5.714 mph
Therefore, the woman's rate for the first part of the journey (on the road running north) is approximately 5.714 mph.