Markus jogs 4 mi around the track at an average rate of r mi/h. Then he cools down by walking 1 mi at a rate 3 mi/h slower. His whole workout last 1 h. At what rates did Markus jog and walk?
Since time = distance/speed,
4/r + 1/(r-3) = 1
r = 6
To solve this problem, we can use the formula:
distance = rate * time
Let's break down the given information:
- Markus jogs 4 miles around the track at an average rate of r mi/h.
- Then he cools down by walking 1 mile at a rate 3 mi/h slower.
Let's assign variables to the rates of jogging and walking:
- Let's call the rate of jogging "j".
- The rate of walking will be "w".
We also know the time for jogging and walking combined is 1 hour.
Now, let's form equations based on the given information:
1. For jogging:
distance_jogging = j * time_jogging
distance_jogging = 4 miles
time_jogging = distance_jogging / rate_jogging
time_jogging = 4 miles / r mi/h
time_jogging = 4/r hours
2. For walking:
distance_walking = w * time_walking
distance_walking = 1 mile
time_walking = distance_walking / rate_walking
time_walking = 1 mile / (r - 3) mi/h
time_walking = 1/(r - 3) hours
Based on the given information, the total workout time is 1 hour, which means the sum of jogging time and walking time should be 1 hour:
time_jogging + time_walking = 1/r + 1/(r - 3) = 1
Now, let's solve the equation:
1/r + 1/(r - 3) = 1
Multiplying through by r(r - 3), we get:
(r - 3) + r = r(r - 3)
2r - 3 = r^2 - 3r
Rearranging and setting the equation to zero:
r^2 - 5r + 3 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
After solving for r, you will find two possible values. Plug each value into the equation (1/r + 1/(r - 3) = 1) to find the corresponding rates of jogging and walking.
Note: Make sure to check if the rates are reasonable (positive values) and consistent with the given information (e.g., faster jogging rate than walking rate).