Question

tylers walk
tyler was at the amusement park. he walked at a steady pace from the ticket booth to the bumper cars.

1. the point on the graph shows his arrival at the bumper cars. what do the coordinates of the point tell us about the situation?
2. the table representing tylers walk shows other values of time and distance. complete the table. next, plot the pairs of values on the grid.
3. what does the point (0,0) mean in this situation?
4. how far away from the ticket booth was tyler after 1 second? label the point on the graph that shows this information in the table with its coordinates.
5. what is the constant of proportionality for the relationship between time and distance?
6. what does it tell you about tylers walk? where do you see it in the graph?

Explanation:

Step1: Determine the meaning of coordinates

In a graph with time (x - axis) and distance (y - axis), for a point \((x,y)\), \(x\) represents the time value and \(y\) represents the distance value.

Step2: Analyze the point \((40,50)\)

The \(x\) - coordinate is \(40\) (time in seconds) and the \(y\) - coordinate is \(50\) (distance in meters). So, it means that after \(40\) seconds, Tyler is \(50\) meters from the ticket booth.

Step3: Analyze the point \((0,0)\)

When \(x = 0\) (time \(t=0\) seconds) and \(y = 0\) (distance \(d = 0\) meters), it means that at time \(t = 0\) (the start), Tyler is at the ticket booth (distance from the ticket booth is \(0\) meters).

Step4: Find the distance after \(1\) second

The constant of proportionality \(k\) (speed) is found by the formula \(k=\frac{y}{x}\). Using the point \((40,50)\), \(k=\frac{50}{40}=\frac{5}{4}=1.25\) m/s. For \(x = 1\) second, \(y=k\times x\). Substituting \(x = 1\) and \(k=\frac{5}{4}\), we get \(y=\frac{5}{4}=1.25\) meters.

Step5: Find the constant of proportionality

The constant of proportionality \(k\) (also known as the unit rate) for a proportional relationship \(y = kx\) (where \(y\) is distance and \(x\) is time) is calculated as \(k=\frac{y}{x}\). Using the point \((40,50)\), \(k=\frac{50}{40}=\frac{5}{4}=1.25\).

Step6: Interpret the constant of proportionality

The constant of proportionality \(k = 1.25\) means that Tyler's speed is \(1.25\) meters per second. In the graph, the slope of the line (since \(y=kx\) is a straight - line equation passing through the origin) represents the constant of proportionality. The steeper the line (for a positive slope), the higher the speed. Here, the slope \(m = k=\frac{5}{4}=1.25\).

Answer:

1. After \(40\) seconds, Tyler is \(50\) meters from the ticket booth.
2. (Since the process of filling the table and plotting is more of a graphical task, but for the distance at \(x = 1\) second, we have \(y = 1.25\) meters. To complete the table, if \(x = 10\), \(y=k\times10=\frac{5}{4}\times10 = 12.5\); if \(x=50\), \(y=\frac{5}{4}\times50 = 62.5\) etc.)
3. At time \(t = 0\) (the start), Tyler is at the ticket booth (distance from the ticket booth is \(0\) meters).
4. \(1.25\) meters, and the point is \((1,1.25)\).
5. \(\frac{5}{4}=1.25\).
6. Tyler's speed is \(1.25\) meters per second. In the graph, it is represented by the slope of the line \(y = 1.25x\) (the line passing through the origin \((0,0)\) and \((40,50)\) etc.).