Question

Many people enjoy challenging themselves by running in marathons. A marathon is a 26-mile race. The runners wear numbers that identify them so their progress can be tracked at checkpoints throughout the race. Such races can be modeled using equations.
An athlete is running in a marathon that eventually finishes at a local high school. After x minutes, the distance the athlete is from the school is given by y (in miles), where x + 10y = 260 (Note: 0 ≤ x ≤ 10 and 0 ≤ y ≤ 26). Complete the steps below to understand how to find the intercepts of the equation and what they represent.
Part 1:Use the relationship between x and y to find the time it takes for the runner to reach the school.
Part2:Use the relationship between x and y to find the distance between the runner and the high school at the start line.
Part3:If the runner ran for 40 minutes from the start line, what is the remaining distance between the runner and the school?
Part4:How much time will the runner take to complete half the distance between the start line and the high school?
Part5:What do the x- and y-intercepts represent in this scenario?

Answer 1

Answer:

We can start by rewriting the initial equation x +10y=260 in order to express how y depends on x.

This is a linear function ( it's a straight line). Linear functions have two intercepts, y and x.

To find x-intercept (also called zero of a function) we set y=0 and we solve for x.

This means that at x=260 our function has a value of 0. y(x) represents a distance from the school. This means that when y(x) is 0 our athlete reached school (finished the race), so this also answers your second question. The time it takes an athlete to finish the race is 260 minutes.

The y-intercept a value of our function at x=0. This gives us a distance of our athlete from the school at the start of the race. We find this by plugin x=0 in our equation. We get that y=26 when x=0. This makes perfect sense. Our athlete is 26 miles away from high school at the start of the race.

Step-by-step explanation: