Question

the clay family is taking a road trip from philadelphia to visit family in nashville. they are averaging 55 miles per hour. after 4 hours, they have 605 miles to go.
a) write an equation to model the miles left to travel, y, after x hours.
b) what does the slope & y - intercept mean in terms of the problem?
c) approximate their total travel time.

Explanation:

Step1: Find the total distance

The distance covered in 4 hours is \(d = 55\times4=220\) miles.
The total distance \(D=220 + 605=825\) miles.

Step2: Write the linear equation

The general form of a linear equation is \(y=mx + b\). Here, the rate of change (slope) \(m=- 55\) (negative because the miles left decrease as time increases) and \(b = 825\).
So the equation is \(y=-55x + 825\).

Step3: Interpret slope and y - intercept

• Slope: The slope \(m=-55\) means that the number of miles left to travel decreases by 55 miles per hour.
• Y - intercept: When \(x = 0\) (at the start of the trip), \(y=825\). So the y - intercept \(b = 825\) is the total distance of the trip.

Step4: Find total travel time

When they reach the destination, \(y = 0\).
Set \(y=-55x + 825=0\).
Solve for \(x\):

$$
LATEXBLOCK0
$$

Answer:

a) \(y=-55x + 825\)
b) Slope: Miles left decrease by 55 per hour. Y - intercept: Total trip distance (825 miles).
c) 15 hours