Question

Mrs. Mitchell, the P.E. teacher, is pairing off students to race against each other. Leah can run 5 yards per second, and Rafi can run 8 yards per second. Mrs. Mitchell decides to give Leah a head start of 24 yards since she runs more slowly. Once the students start running, Rafi will quickly catch up to Leah. How far will Rafi have to run? How long will that take?

Answer 1

Rafi and Leah will neet at the 64 yard point after 8 seconds. [They say hello and decide to take a break.]

Explanation:

Let L and R be the speeds for Leah and Rafi, respectively.

The distance travelled for each for a time, x, is given by the expressions:

Leah: x*L (where x is in seconds and L is yards/sec)

Rafi: x*R (where x is in seconds and R is yards/sec)

[Note that the time, x, will be the same for both Leah and Rafi. x is the time Rafi catches up after both start the race at the same time.

The distance for Rafi is x*R.

The distance for Leah, however, is x*L PLUS the 24 yard head start

Let D be the distance at which Rafi catches up to Leah.

For Leah, the total distance would be (x*L+24 yards)

Leah is being given a 24 yard head start

When they meet, their distrances from the start line are the same. So we can write:

x*R = (x*L+24 yards)

Inserting the given values for R and L:

x*(8 yards/sec) = (x*(5 yards/sec)+24 yards)

Since x is in seconds, the terms with x in them will reduce to just yards. Lets implfy by removing units for a step or two and then adding yards to the final answer:

x*(8) = (x*(5)+24)

8x = 5x + 24

3x = 24

x = 8 seconds

Rafi will catch up to Leah in 8 seconds.

Distance:

Rafi: (8 sec)*(8 yrds/sec) = 64 yards

Leah: (8 sec)*(5 yrds/sec) = 40 yards

Since Leah had a 24 yard head start, she would be at the 64 yard point in 8 seconds. They meet at 64 yards.