You jog at 6.0 mi/h for 5.0 mi, then you jump into a car and drive for another 5.0 mi. With what average speed must you drive if your average speed for the entire 10.0 miles is to be 10.0 mi/h?
To find the average speed for the entire 10.0 miles, we can use the formula:
Average Speed = Total Distance / Total Time
First, let's find the time it took to jog for 5.0 miles at a speed of 6.0 mi/h.
Time = Distance / Speed
Time = 5.0 mi / 6.0 mi/h
Time = 0.833 hours
Next, we need to find the time it took to drive for 5.0 miles at an unknown speed, which we will call "x mi/h".
Time = Distance / Speed
Time = 5.0 mi / x mi/h
Time = 5.0 / x hours
Now, we can set up an equation for the total time:
Total Time = 0.833 hours + 5.0 / x hours
We are given that the average speed for the entire 10.0 miles is to be 10.0 mi/h. So we can set up another equation for the average speed:
Average Speed = Total Distance / Total Time
10.0 mi/h = 10.0 mi / (0.833 hours + 5.0 / x hours)
To find x, we can solve this equation:
10.0 mi/h = 10.0 mi / (0.833 hours + 5.0 / x hours)
To simplify, we can multiply both sides of the equation by the denominator:
10.0 mi/h * (0.833 hours + 5.0 / x hours) = 10.0 mi
Now, we can distribute the 10.0 mi/h:
8.33 hours + 50.0 / x hours = 10.0 mi
Next, let's isolate the term with x on one side of the equation:
50.0 / x hours = 10.0 mi - 8.33 hours
Now, subtract 8.33 hours from 10.0 mi:
50.0 / x hours = 1.67 hours
To solve for x, we can divide both sides of the equation by 1.67 hours:
50.0 / x = 1.67
Finally, we can solve for x by multiplying both sides of the equation by x:
50.0 = 1.67 * x
Dividing both sides by 1.67:
x = 50.0 / 1.67
x ≈ 29.94
Therefore, to have an average speed of 10.0 mi/h for the entire 10.0 miles, you must drive at an average speed of approximately 29.94 mi/h for the second 5.0 miles.