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.

Question

Use this information to write a function f(x) for the expected response time to a neighborhood that is x miles away.

Answer 1

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.

Answer 2

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

Answer 3

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.