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 hour. At what rates did Markus jog and walk?
To solve this problem, we can use the formula:
Time = Distance / Rate
Let's assume that Markus jogs at a rate of "r" mi/h. We are given that he jogs 4 miles, so the time it takes him to jog 4 miles is:
Time for jogging = 4 / r
Now, during his cool-down, Markus walks at a rate that is 3 mi/h slower than his jogging rate. We can represent his walking rate as (r - 3) mi/h. We are given that he walks 1 mile, so the time it takes him to walk 1 mile is:
Time for walking = 1 / (r - 3)
We are also given that the whole workout lasts 1 hour. Therefore, the total time for jogging and walking is:
Total time = Time for jogging + Time for walking
1 = 4 / r + 1 / (r - 3)
Now, let's simplify this equation:
(r(r - 3)) = (4(r - 3)) + 4r
r^2 - 3r = 4r - 12 + 4r
r^2 - 7r - 12 = 0
We can solve this quadratic equation using factoring or the quadratic formula. Factoring this equation, we have:
(r - 4)(r + 3) = 0
Setting each factor equal to zero, we get two possible solutions:
r - 4 = 0 or r + 3 = 0
Solving for "r," we find:
r = 4 or r = -3
Since we are looking for a positive rate, we can disregard the solution r = -3.
Therefore, Markus jogs at a rate of 4 mi/h, and he walks at a rate of (4 - 3) = 1 mi/h.