Question

the linear function f computes the distance y in miles between a car and a city after x hours, where $0\leq x\leq10$. the graphs of f and the horizontal lines $y = 200$ and $y = 300$ are shown. use the graph to answer the following.

1. is the car moving toward or away from the city?

$circ$ away
$circ$ toward

1. determine the time when the car is 300 miles from the city.
2. when is the car from 200 to 300 miles (inclusively) from the city?

Explanation:

Step1: Analyze direction of movement

As \(x\) (time) increases, \(y=f(x)\) (distance) increases from 100 miles to over 400 miles. So the car moves away.

Step2: Find time for \(y=300\)

First, find the equation of \(y=f(x)\). The line passes through \((0,100)\) and \((10,500)\). Slope \(m=\frac{500-100}{10-0}=40\). Equation: \(y=40x+100\). Set \(y=300\):
$$300=40x+100$$
Solve for \(x\):
$$40x=300-100=200$$
$$x=\frac{200}{40}=5$$

Step3: Find time range for 200-300 miles

Set \(y=200\):
$$200=40x+100$$
$$40x=100$$
$$x=\frac{100}{40}=2.5$$
From Step2, \(y=300\) at \(x=5\). So the range is \(2.5\leq x\leq5\).

Answer:

1. Away
2. 5 hours
3. Between 2.5 hours and 5 hours (inclusively)