Find the equations.

Question

Princess Peach is racing with all the MarioKart characters. I know that at 3 seconds she is 25 meters into the race and at 9 seconds she is 61 meters.

Bowser is also racing. Bowser travels at 5 meters per second. Bowser had an unknown headstart, but I don't know what it was. I do know that Bowser was 80 meters after 10 seconds.

a)Find the equations of both lines.

b)Who won the race if the race was 90 meters long. Prove it using your equations.

Answer 1

Because the question explicitly states that you are graphing lines, it is a good idea to go into this thinking about what each value represents. In the question, we talk about time, distance, head start, and speed. Each of these represents a concept you are already familiar with regarding lines.

Remember that the format for linear equations is:

y = mx + b.

In this problem, y is distance from the starting line, x is time that has elapsed, b is head start from the starting line, and m is the speed a character is traveling.

To start with, let's examine Princess Peach's speed. We are given two points to use when creating her equation: Point A is (3,25) and Point B is (9,61).

Remember that we can finding slope (m) using two points on a line with the following formula:

m = (y₂ - y₁)/(x₂ - x₁)

Using the formula for slope, we can determine that m = (61 - 25)/(9 - 3) = 36/6 = 6

We also know that Princess Peach started at the starting line with no head start, so the y-intercept will be 0 (b=0).

Princess Peach has a linear equation of:

y = 6x + 0 or y = 6x

Next we will work on Bowser's equation. We know that he is traveling at 5 m/s. Remember that we said that slope represents speed. So we know that m = 5. Bowser has an unknown head start, so b = ? for now. We do know that after 10 seconds, Bowser was 80 meters from the start line. With those numbers, we can solve for b. Start with your linear formula:

y = mx + b

substitute what you know

80 = (5)(10) + b

and solve for b

80 = 50 + b

30 = b

Bowser has a linear equation of:

y = 5x + 30

Now we just compare the two to see who will win:

y = 5x + 30

y = 6x + 0

We are racing to 90 meters, so use that for your y value (which represents distance) and solve for x (time)! The x value you get for each racer represents the time that it takes them to reach 90 meters!

Peach:

90 = 6x + 0

90 = 6x

90/6 = x

15 seconds = x

Bowser:

90 = 5x + 30

60 = 5x

60/5 = x

12 seconds = x

It looks like Bowser won the race! He reached 90 meters a full 3 seconds before Princess Peach.