diana drove 320 miles, then increased her speed by 10 miles per hour for 500 miles. if the second part of the trip took 2 hours longer than the first, find her average speed.

let her initial speed be x mph

time for first trip = 320/x

new speed = x+10 mph
time for second part of trip = 500/(x+10)
so..

500/(x+10) - 320/x = 2
times x(x+10)
500x - 320(x+10) = 2x(x+10)
500x - 320x - 3200 = 2x^2 + 20x
2x^2 - 160x + 3200 = 0
x^2 - 80x + 1600 = 0
(x-40^2 = 0
x = 40

time for first leg = 320/40 =8 hrs
time for 2nd leg = 500/50 = 10 hrs
avg speed = total distance/total time
= 820/18 = appr 45.56 mph

To find Diana's average speed, we need to calculate her total time traveled and divide it by the total distance covered.

Let's break down the problem step by step:

Step 1: Calculate the time Diana spent on the first part of the trip.
We know the distance covered on the first part of the trip is 320 miles. However, we don't know the speed, so let's call it "x" (miles per hour). To calculate time, we can use the formula: time = distance / speed.
So, the time for the first part of the trip is t1 = 320 / x.

Step 2: Calculate the time Diana spent on the second part of the trip.
We know the distance covered on the second part of the trip is 500 miles. Diana's speed on this part of the trip increased by 10 miles per hour compared to the first part, so her speed would be (x + 10) miles per hour. We also know that the second part of the trip took 2 hours longer than the first part. Therefore, we can calculate the time for the second part of the trip as t2 = 500 / (x + 10) and t2 = t1 + 2.

Step 3: Solve the equation t1 + 2 = 500 / (x + 10) to find the value of x.
First, we multiply both sides of the equation by (x + 10) to eliminate the fraction. This gives us:
(t1 + 2) * (x + 10) = 500
Expanding and rearranging the equation, we get:
x(t1 + 10) + 2(x + 10) = 500
Now, substitute the value of t1 = 320 / x into the equation:
x(320 / x + 10) + 2(x + 10) = 500
Simplifying further:
320 + 10x + 2x + 20 = 500
12x + 340 = 500
12x = 160
x = 160 / 12
x ≈ 13.33 (rounded to two decimal places)

Step 4: Calculate the total time traveled.
Now that we have the value of x (Diana's speed on the first part of the trip), we can calculate the total time traveled. It is the sum of t1 and t2:
Total time traveled = t1 + t2 = 320 / x + 500 / (x + 10)

Step 5: Calculate the average speed.
Average speed = Total distance / Total time traveled
To find the average speed, we add the distances on the first and second parts of the trip:
Total distance = 320 + 500

Finally, we divide the total distance by the total time traveled to find the average speed:
Average speed = (320 + 500) / (320 / x + 500 / (x + 10))

Substituting the value of x ≈ 13.33 into this equation will give us the average speed.