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 determine the rates at which Markus jogged and walked, we need to set up a system of equations using the given information.
Let's represent the rate at which Markus jogs as r mi/h.
Given that Markus jogs 4 miles at an average rate of r mi/h, we can write the equation:
Jogging time = Distance / Rate
4 mi / r mi/h = 4/r h
Next, we need to determine the rate at which Markus walks during the cooldown.
Given that he cools down by walking 1 mile at a rate 3 mi/h slower, we can write the equation:
Walking time = Distance / Rate
1 mi / (r - 3) mi/h = 1/(r - 3) h
Since the total workout duration is 1 hour, the sum of jogging and walking times must equal 1:
Jogging time + Walking time = 1
4/r + 1/(r - 3) = 1
Let's solve this equation to find the values of r and r - 3.
Multiply both sides of the equation by r(r - 3) to eliminate the denominators:
4(r - 3) + r = r(r - 3)
Simplify the equation:
4r - 12 + r = r^2 - 3r
Combine like terms:
5r - 12 = r^2 - 3r
Rearrange the equation to form a quadratic equation:
r^2 - 8r + 12 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula. Factoring, we get:
(r - 2)(r - 6) = 0
Setting each factor to zero, we get two possible values for r:
r - 2 = 0 --> r = 2
r - 6 = 0 --> r = 6
Since the rate represents speed, we can discard the negative value of r.
Therefore, Markus jogged at a rate of 6 mi/h and walked at a rate of 6 - 3 = 3 mi/h.