A fire department has an average response time of 5 minutes for a neighborhood 8 miles away from the station and a response time of 6.5 minutes for a neighborhood 12 miles from the station.
Use this information to write a function f(x) for the expected response time to a neighborhood that is x miles away.
We can assume that the relationship between distance and response time is linear, so we can use the slope-intercept formula:
f(x) = mx + b
where m is the slope (rate of change in response time per mile) and b is the y-intercept (the expected response time for a neighborhood at 0 miles away, which we assume to be 0).
To find the slope, we can use the two data points given:
m = (6.5 - 5) / (12 - 8) = 0.375
(Note that the units cancel out, as we are dividing minutes by miles.)
So the function for expected response time is:
f(x) = 0.375x
This means that for every mile farther away a neighborhood is from the station, the expected response time increases by 0.375 minutes, or 22.5 seconds.
Two input-output pairs for function f(x) are (0,5) and (10, 35).
Two input-output pairs for function g(x) are (-2, 3) and (6, 20) .
Paige says that function f(x) has a steeper slope. Explain whether Paige is correct or not. Use evidence from your own calculations
To determine the slope of each function, we can use the slope-intercept formula:
f(x) = mx + b
g(x) = nx + c
where m and n are the slopes of each function, and b and c are the y-intercepts.
Using the given input-output pairs for f(x), we can solve for the slope:
m = (35 - 5) / (10 - 0) = 3
Using the given input-output pairs for g(x), we can solve for the slope:
n = (20 - 3) / (6 - (-2)) = 17/8 = 2.125
We can see that function g(x) actually has a steeper slope than function f(x). Thus, Paige's statement is incorrect.
To write the function f(x) for the expected response time to a neighborhood that is x miles away, we can make use of the given data points.
Here, we have two data points:
- A neighborhood 8 miles away with an average response time of 5 minutes.
- A neighborhood 12 miles away with an average response time of 6.5 minutes.
To find the relationship between the distance and response time, we can use a linear equation since the response time seems to increase linearly with the distance.
Let's start by finding the slope of the line using the two data points:
Slope = (Change in Response Time) / (Change in Distance)
Slope = (6.5 minutes - 5 minutes) / (12 miles - 8 miles)
= 1.5 minutes / 4 miles
= 0.375 minutes/mile
Now, we can plug in one of the data points into the equation y = mx + b, where y represents the response time, x represents the distance, m represents the slope, and b represents the y-intercept.
Using the data point (8 miles, 5 minutes):
5 minutes = 0.375 minutes/mile * 8 miles + b
Simplifying this equation:
5 minutes = 3 minutes + b
Subtracting 3 minutes from both sides:
2 minutes = b
Now we have the y-intercept (b = 2 minutes), and we can write the equation for the expected response time to a neighborhood x miles away:
f(x) = 0.375 minutes/mile * x miles + 2 minutes
Therefore, the function for the expected response time to a neighborhood that is x miles away is:
f(x) = 0.375x + 2