Question

error analysis a question on a test asks students to find the speed at which a car travels. the graph shows a proportional relationship between the distance traveled in miles and time in hours. anna incorrectly says that the speed of the car is \\(\frac{1}{55}\\) mile per hour. what is the speed of the car? what error might anna have made? a. she used \\(\frac{\text{change in } y\text{-coordinates}}{\text{change in } x\text{-coordinates}}\\) to find a unit rate, not \\(\frac{\text{change in } x\text{-coordinates}}{\text{change in } y\text{-coordinates}}\\) b. she used \\(\frac{\text{change in } x\text{-coordinates}}{\text{change in } y\text{-coordinates}}\\) to find a unit rate, not \\(\frac{\text{change in } y\text{-coordinates}}{\text{change in } x\text{-coordinates}}\\)

Explanation:

Step1: Recall the formula for speed

Speed is calculated as the ratio of distance (y - coordinate) to time (x - coordinate) in a proportional relationship, i.e., \( \text{Speed}=\frac{\text{Change in }y}{\text{Change in }x} \).
From the graph, when \( x = 1 \) hour, \( y=55 \) miles (we can see from the point (1, 55) on the line).

Step2: Calculate the correct speed

Using the formula \( \text{Speed}=\frac{\text{Distance}}{\text{Time}}=\frac{\text{Change in }y}{\text{Change in }x} \). Taking the point (1, 55), the change in \( y \) is 55 miles and change in \( x \) is 1 hour. So speed \(=\frac{55}{1} = 55\) miles per hour.

Step3: Analyze Anna's error

Anna got \( \frac{1}{55} \) which is \( \frac{\text{Change in }x}{\text{Change in }y} \) (since \( \frac{1}{55}=\frac{\text{Time}}{\text{Distance}} \)) instead of \( \frac{\text{Change in }y}{\text{Change in }x} \) (which is \( \frac{\text{Distance}}{\text{Time}} \)). So she used \( \frac{\text{change in }x\text{-coordinates}}{\text{change in }y\text{-coordinates}} \) to find the unit rate instead of \( \frac{\text{change in }y\text{-coordinates}}{\text{change in }x\text{-coordinates}} \).

Answer:

The speed of the car is 55 miles per hour. Anna's error was that she used \( \frac{\text{change in }x\text{-coordinates}}{\text{change in }y\text{-coordinates}} \) (i.e., \( \frac{\text{Time}}{\text{Distance}} \)) to find the unit rate instead of \( \frac{\text{change in }y\text{-coordinates}}{\text{change in }x\text{-coordinates}} \) (i.e., \( \frac{\text{Distance}}{\text{Time}} \)).