When training for long-distance events like marathons, athletes need to carefully track their workouts to build endurance safely. Breaking down a workout session into different speeds – walking, jogging, and running – helps runners avoid injury and gradually increase their stamina. Mathematics can help us analyze these mixed-pace workouts to better understand the total distance covered and time spent training.

Rosa is training for a marathon. Every day, she does a combination of walking, jogging, and running. She calls it “doing her cardio.” Her goal is to gradually increase the number of consecutive miles that she can do. Below is an expression for the total time Rosa spent doing her cardio on Tuesday where [latex]j[/latex] is her jogging speed in miles per hour.

[latex]\dfrac{5j^2+4j-4}{j^3-4j}[/latex]

To mix up her training, Rosa decided to spend a week doing a combination of walking, power walking, and sprinting. She walked at a speed that was 1mph less than her power walking speed and sprinted at a speed that was the square of her walking speed. The following expression is the total time Rosa spent doing this cardio combo for the week where [latex]p[/latex] is her power walking speed in miles per hour.

Another week, Rosa decided to go on a run each morning that would start with a set distance at a warm-up speed, [latex]w[/latex], in miles per hour, and then finish at a faster speed for a distance that varied with her warm-up pace. The following expression represents the total time for each morning run.