Markus jogs 4 mi around a 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 lasts 1 h. At what rates did Markus jog and walk?

To solve this problem, we can use the information given to create two equations based on the distances Markus jogs and walks and the rates at which he jogs and walks. Let's solve it step by step:

Let's assume Markus' jogging rate is represented by r mi/h. Since he jogs for 4 mi, we can calculate the time it takes for him to complete this distance using the formula:

Time = Distance / Rate

So, the time it takes for Markus to jog 4 mi is:

Time jogging = 4 mi / r mi/h

Now, let's consider his cool-down walk. We know that the distance of his walk is 1 mi, and it is done at a rate 3 mi/h slower than his jogging rate. This means his walking rate is (r - 3) mi/h. Using the same formula as before, we can calculate the time it takes for Markus to complete this distance:

Time walking = 1 mi / (r - 3) mi/h

According to the problem, the total time Markus spent on his workout is 1 hour. So, the sum of the time jogging and the time walking should equal 1:

Time jogging + Time walking = 1

Substituting the expressions we found for the times:

4 mi / r mi/h + 1 mi / (r - 3) mi/h = 1

Now, we can solve this equation to find the value of r, which represents Markus' jogging rate. Let's multiply all terms by the least common multiple of the denominators (r and r - 3) to eliminate the fractions:

4(r - 3) + r = r(r - 3)

Expanding and simplifying this equation gives us:

4r - 12 + r = r^2 - 3r

Combining like terms:

5r - 12 = r^2 - 3r

Rearranging to set the equation equal to zero:

r^2 - 8r + 12 = 0

Now, we can use the quadratic formula to solve for r:

r = (-b ± √(b^2 - 4ac)) / 2a

For this equation, a = 1, b = -8, and c = 12. Plugging in these values:

r = (-(-8) ± √((-8)^2 - 4(1)(12))) / (2(1))

Simplifying:

r = (8 ± √(64 - 48)) / 2
r = (8 ± √16) / 2

Simplifying further:

r = (8 ± 4) / 2

This gives us two possible values for r:

1. r = (8 + 4) / 2 = 12 / 2 = 6
2. r = (8 - 4) / 2 = 4 / 2 = 2

Since the rate of speed cannot be negative, we discard the second solution r = 2. Therefore, the value of r representing Markus' jogging rate is 6 mi/h.

To find his walking rate, we subtract 3 mi/h from his jogging rate:

Walking rate = Jogging rate - 3 = 6 mi/h - 3 mi/h = 3 mi/h

Thus, Markus jogs at a rate of 6 mi/h, and when he cools down, he walks at a rate of 3 mi/h.