Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Mrs. Hanson, the P.E. teacher, is pairing off students to race against each other. Britney can run 4 yards per second, and Darnell can run 5 yards per second. Mrs. Hanson decides to give Britney a head start of 31 yards since she runs more slowly. Once the students start running, Darnell will quickly catch up to Britney. How long will that take? How far will Darnell have to run?

Answer 1

It will take Darnell 31 seconds to catch up to Britney, and he will have to run 155 yards. We solved this by setting up and solving a system of equations representing the distance each runner covers over time.

Step-by-step explanation:

To find out how long it will take Darnell to catch up to Britney and how far he will have to run, we will set up a system of equations that represent their progress in the race. Let t be the time in seconds after they start running, and d be the distance in yards that Darnell runs.

Britney's equation, given her speed of 4 yards per second and a head start of 31 yards, would be:

Distance Britney runs = Speed of Britney × Time + Head start

Britney's distance = 4t + 31

Darnell's equation, with his speed of 5 yards per second and no head start, would be:

Distance Darnell runs = Speed of Darnell × Time

Darnell's distance = 5t

To find when Darnell catches up to Britney, we set their distances equal:

4t + 31 = 5t

By moving all the terms involving t to one side, we get:

t = 31

This means that it will take Darnell 31 seconds to catch up to Britney.

Now, we calculate the distance Darnell travels in 31 seconds:

Darnell's distance = 5 × 31

Darnell's distance = 155 yards.