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