Question

1. check for reasonableness a question on a test provides this graph and asks students to find the speed at which the car travels. anna incorrectly says that the speed of the car is \\(\frac{1}{64}\\) mile per hour.

a. what is the speed of the car?
b. what error might anna have made?
(graph: title: speed of a car, x - axis: time (hours) with labels 0,1,2,3,4; y - axis: distance (miles) with labels 0,64,128,192,256; a line passes through the origin and other points on the grid.)

Explanation:

Part (a)

Step1: Recall speed formula

Speed is calculated as \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). From the graph, we can take a point, for example, when time \( x = 1 \) hour, distance \( y = 64 \) miles (or check the slope of the line since it's a linear graph, slope \( m=\frac{\Delta y}{\Delta x} \) which is speed here).

Step2: Calculate speed

Using the formula, with distance \( d = 64 \) miles and time \( t = 1 \) hour, \( \text{Speed}=\frac{64}{1}=64 \) miles per hour. We can also check other points: at \( x = 2 \) hours, \( y = 128 \) miles, so \( \frac{128}{2}=64 \) mph, confirming the speed.

Anna likely mixed up the variables for distance and time. Instead of using \( \text{Speed}=\frac{\text{Distance}}{\text{Time}} \), she might have calculated \( \frac{\text{Time}}{\text{Distance}} \), leading to \( \frac{1}{64} \) mph. This is a common error of inverting the ratio for speed (confusing the dependent and independent variables on the graph).

Answer:

The speed of the car is 64 miles per hour.

Part (b)