Sam and Alice decide to have another race. Below is the information about each car.

Question

when the timer started Sam car was 1 foot in front of the starting line his car travel 13.5 in feet per second

Alice car Travel 12 feet per second After 3 seconds her Car was 40 Feet in front of the start line

Let x represent the number of seconds into the race

Let y represent the distance from the start line

Part 1: Write an equation to model Sam's car. 

Part 2: Write an equation to model Alice's car.  

3. The race is over at 15 seconds. The winner is defined as the person that is furthest from the start line. Who won the race? How do you know? 

Answer 1

First, draw and label a diagram with the five following points:

1. a line to represent the "starting line"
2. a point a little bit after the starting line with a distance of 1 foot (label as "Sam 0 sec")
3. an arrow extending after "Sam 0 sec" point and label as "13.5 ft/sec"
4. an arrow extending from the starting line and label as "12 feet/sec"
5. a point further from the starting line with a distance of 40 feet (label as "Alice 3 sec")

Part 1. Next, make a table for Sam of x (which represents the number of seconds into the race) and y (which represents the distance from the starting line)

x = 0 seconds y = 1 foot *since the problem gave us that Sam started 1 foot in front of the starting line*

x = 1 second y = 1 second x 13.5 feet/second = 13.5 feet + 1 foot = 14.5 feet

*here we are able to multiply Sam's rate 13.5 ft/sec by the number of seconds (1 sec)

but we need to remember that Sam started at 1 foot so we have to add that

x = 2 seconds y = 2 seconds x 13.5 feet/second = 27 feet + 1 foot = 28 feet

*same as 1 sec applies here; we can multiply the rate by the number of seconds

(2 sec) but need to keep in mind that Sam started 1 foot in front of the starting line

If you notice for 1 and 2 seconds we have just been multiplying x (the number of seconds) by the rate (13.5 ft/sec) and adding 1 foot to that product. Therefore, we can come up with the following model equation for Sam:

y = 13.5x + 1

Part 2. Next, make a table for Alice of x (which represents the number of seconds into the race) and y (which represents the distance from the starting line)

x = 0 seconds y = ? *we do not know this yet since the problem did not provide Alice's starting distance

x = 3 seconds y = 40 feet *we know this since it is given in the problem

We are also given Alice's rate: 12 ft/sec

Let us use Sam's equation as a model

y = 12x + ? *? = Alice's starting position at 0 seconds

Since we do not know Alice's starting position we will need to use the given information at 3 seconds to plug in and solve for "?"

y = 40 ft and x = 3 sec

40 = 12(3) + ?

40 = 36 + ?

40 - 36 = ?

4 feet = ?

Therefore, we have solved that ? = 4 feet and so Alice's model equation is:

y = 12x + 4

3.To figure out who won after 15 seconds, you will basically need to plug in 15 seconds for each of the model equations you found previously.

Sam : y = 13.5x + 1

*plug in 15 for x*

y = 13.5(15) + 1

y = 202.5 feet + 1 foot

y = 203.5 feet

Alice: y = 12x + 4

*plug in 15 for x*

y = 12(15) + 4

y = 180 feet + 4 feet

y = 184 feet

Since 203.5 feet is bigger than 184 feet, Sam won the race.