We are given a 50 foot head start to a friend in a race. The distance y in feet our friend runs after x seconds is represented by the given linear function.
y=14x+50
Our distance at various times throughout the race is showing in the given table.
Time (seconds), x
2
4
6
8
Distance (feet), y
38
76
114
152
Let's write a linear function that represents our distance. To do so, recall that functions written in slope-intercept form follow a specific format.
y= mx+ b
In this form, m is the slope and b is the y-intercept. We will identify these values using the given table. Let's start by substituting the points (2,38) and (4,76), into the slope formula and calculating m.
We calculated that the slope is 19. Now, remember that the y-intercept is the y-value where the line crosses the y-axis. It occurs when x=0. In our case, x=0 represents the beginning of the race. Then, the y-intercept will be 0 because the distance traveled at the beginning of the race is equal to 0.
y= 19x+ 0 → y=19x
We want to calculate for what distance we will win the race. This will occurs when the distance traveled by our friend is less than the distance traveled by us. This can be represented by the following inequality.
19x>14x+50
Let's solve this equation for x to find when this inequality is true.
This means that after 10 seconds, our distance traveled will be greater than the distance traveled by our friend. Now, let's substitute x= 10 into the obtained function for our distance.
