Question

The graph shows the relationship between the distance a professional biker is traveling and the time in hours.

A graph is shown in the xy-plane. The x-axis is labeled as Time in minutes and the y-axis is labeled as Distance in miles. The values on the x-axis ranges from 0 to 10 in increments of 1 and the values on the y-axis ranges from 0 to 4 in increments of 0 decimal point 5. A line starts from the origin and goes up. Five points are marked on the line. For 1 on the x-axis the value of y is just below 0 decimal point 5 on the line, for 2 on the x-axis the value of y is just above 0 decimal point 5 on the line, for 5 on the x-axis the value of y is between 1 decimal point 5 and 2 on the line, for 8 on the x-axis the value of y is just below 3 on the line, and for 10 on the x-axis the value of y is 3 decimal point 5.

Define variables and write an equation to represent the relationship between the quantities.
How far would the biker travel in 20 minutes?
If the biker traveled 48 miles, how many minutes did he bike? Round to the nearest tenth.

Answer 1

Let x represent the time in minutes the biker travels, and let y represent the distance, in miles, the biker travels. The relationship between the quantities is linear, as shown in the graph.

We can use two points on the graph to find the equation of the line: (2, just above 0.5) and (5, between 1.5 and 2). Let's estimate the distance at these points to be 0.6 miles and 1.7 miles, respectively.

First, let's find the slope (m) of the line:
m = (y2 - y1) / (x2 - x1) = (1.7 - 0.6) / (5 - 2) = 1.1 / 3 = 0.3667

Now let's find the y-intercept (b) of the line:
y = mx + b
Since we know one point on the line (2, 0.6), we can substitute it into the equation to find the y-intercept:
0.6 = 0.3667(2) + b
0.6 = 0.7334 + b
b = -0.1334

So the equation representing the relationship between the quantities is:
y = 0.3667x - 0.1334

To find how far the biker would travel in 20 minutes, substitute 20 for x:
y = 0.3667(20) - 0.1334
y = 7.334 - 0.1334
y ≈ 7.2 miles

If the biker traveled 48 miles, we can use the equation to find out how many minutes he biked. Substitute 48 for y and solve for x:
48 = 0.3667x - 0.1334
48.1334 = 0.3667x
x ≈ 131.3 minutes

The biker would travel about 7.2 miles in 20 minutes, and if he traveled 48 miles, he biked for about 131.3 minutes.