Question

You and a friend race each other. You give your friend a 50-foot head start. The distance y (in feet) your friend runs after x seconds is represented by the linear function y=14x+50 . The table shows the distances you run.


Time (seconds) x : 2, 4, 6, 8

Distinct (feet) y : 38, 76, 114, 152


A. what is the rate per second.


B. Write a linear function that relates your distance to the number of seconds. In the same coordinate plane, graph the linear functions that represent the distances of you and your friend.

Answer 1

A) what is the rate per second

From the table Δy / Δx = 38/2 which is 19
The equation for me will thus be y=19x + 0
the equation for my friend is given as y = 14x + 50
The rate per second for me is 19 ft/sec and my friend's is 14 ft/sec.

B) the linear equations for this question are y = 19x for me and y= 14x + 50 for my friend.

Answer 2

Answer and Explanation :

Given : You and a friend race each other. You give your friend a 50-foot head start. The distance y (in feet) your friend runs after x seconds is represented by the linear function$y=14x+50$ . The table shows the distances you run.

Time (seconds) x : 2, 4, 6, 8

Distinct (feet) y : 38, 76, 114, 152            

To find :

Part A - What is the rate per second?

Solution :

Friend distance equation is given by, $y=14x+50$

The rate is defined as the slope of the line.

The rate per second of friend equation is 14 feet/second.

Now, To determine the rate per second of mine is

Time (seconds) x : 2,  4,   6,    8

Distinct (feet)    y : 38, 76, 114, 152    

We have seen that  the value of x and y changes at same rate.

So, The rate per second is the slope between two points.

$m=\frac{76-38}{4-2}$  

$m=\frac{38}{2}$        

$m=16$      

The rate per second of mine is 16 feet/second.        

Part B - Write a linear function that relates your distance to the number of seconds. In the same coordinate plane, graph the linear functions that represent the distances of you and your friend.          

We know, the slope of mine equation is  m=16

Now, substitute one point (2,38) and slope (m=16) in point-slope form.

$y-38=16(x-2)$      

$y=16(x-2)+38$      

$y=16x-32+38$  

$y=16x+6$  

The linear function that relates mine distance to the number of seconds is $y=16x+6$    

Now, Plot both equations on the graph.

Equation 1 - Friend's equation $y=14x+50$ shown by red line.

Equation 2 - Mine equation $y=16x+6$ shown by blue line.

Refer the attached figure below.