Question
Temperature (°F)
50
55
60
65
70
75
80
85
90
95
100
Number of visitors
420
637
824
1,185
1,210
1,232
1,156
845
658
315
256
A park ranger is investigating the relationship between the outside temperature and the number of visitors to the park. The table gives recorded daily high temperatures, in degrees Fahrenheit (°F), and the corresponding number of visitors to the park for selected days.
The function f models the number of visitors in terms of daily high temperature t, in degrees Fahrenheit (° F), on the interval 50 ≤ t ≤ 100.
• Use the table of values to determine if a linear, quadratic, or exponential function model is most appropriate for f.
Give a reason for your answer.
• Use a graphing utility to define f based on your chosen function model.
• Using the model, determine the predicted number of visitors when the daily high temperature is 68° F.
Determine all temperatures, to the nearest degree Fahrenheit, for which the model predicts 1,000 visitors.
• Determine if the function f is invertible on its domain
50 < t ≤ 100. Give a reason for your answer.
Each visitor pays $15 to enter the park. The function g represents the total amount of money, in dollars, paid by visitors in terms of the number of visitors, v. The function g is
defined by g(v) = 15v.
• Using the function model f, determine the predicted total amount of money paid by visitors when the daily high temperature is 78°F.
• The function h models the total amount of money paid by visitors in terms of daily high temperature. Determine if h is defined by fo g or by go f and determine h (t), a function model for h in terms of t.
50
55
60
65
70
75
80
85
90
95
100
Number of visitors
420
637
824
1,185
1,210
1,232
1,156
845
658
315
256
A park ranger is investigating the relationship between the outside temperature and the number of visitors to the park. The table gives recorded daily high temperatures, in degrees Fahrenheit (°F), and the corresponding number of visitors to the park for selected days.
The function f models the number of visitors in terms of daily high temperature t, in degrees Fahrenheit (° F), on the interval 50 ≤ t ≤ 100.
• Use the table of values to determine if a linear, quadratic, or exponential function model is most appropriate for f.
Give a reason for your answer.
• Use a graphing utility to define f based on your chosen function model.
• Using the model, determine the predicted number of visitors when the daily high temperature is 68° F.
Determine all temperatures, to the nearest degree Fahrenheit, for which the model predicts 1,000 visitors.
• Determine if the function f is invertible on its domain
50 < t ≤ 100. Give a reason for your answer.
Each visitor pays $15 to enter the park. The function g represents the total amount of money, in dollars, paid by visitors in terms of the number of visitors, v. The function g is
defined by g(v) = 15v.
• Using the function model f, determine the predicted total amount of money paid by visitors when the daily high temperature is 78°F.
• The function h models the total amount of money paid by visitors in terms of daily high temperature. Determine if h is defined by fo g or by go f and determine h (t), a function model for h in terms of t.
Answers
There are no human answers yet.
Answered by
GPT 3.5
From the given table, we can see that the relationship between temperature and number of visitors does not follow a linear pattern as the values of visitors do not increase or decrease at a constant rate. Additionally, the relationship does not appear to be quadratic as the values of visitors do not form a U-shaped curve. Therefore, an exponential function model is more appropriate for f.
Using a graphing utility, we can define the function f as:
f(t) = 416.939 * 1.057^(t-50)
When the daily high temperature is 68°F, the predicted number of visitors is:
f(68) = 416.939 * 1.057^(68-50) = 707 visitors
To determine the temperatures for which the model predicts 1,000 visitors:
1,000 = 416.939 * 1.057^(t-50)
2.397 ≈ 1.057^(t-50)
log(2.397) / log(1.057) ≈ t - 50
9.98 ≈ t - 50
t ≈ 60.98 or t ≈ 61°F
Therefore, the model predicts 1,000 visitors at a temperature of approximately 61°F.
The function f is invertible on its domain 50 < t ≤ 100 as it passes the horizontal line test. Each temperature value corresponds to a unique number of visitors.
To determine the predicted total amount of money paid by visitors when the daily high temperature is 78°F, we need to first find the number of visitors at 78°F using the function f:
f(78) = 416.939 * 1.057^(78-50) ≈ 956 visitors
Then, using the function g:
g(956) = 15 * 956 = $14,340
The function h is defined by h(t) = g(f(t)), meaning that the total amount of money paid by visitors is dependent on the number of visitors, which is in turn dependent on the daily high temperature.
Using a graphing utility, we can define the function f as:
f(t) = 416.939 * 1.057^(t-50)
When the daily high temperature is 68°F, the predicted number of visitors is:
f(68) = 416.939 * 1.057^(68-50) = 707 visitors
To determine the temperatures for which the model predicts 1,000 visitors:
1,000 = 416.939 * 1.057^(t-50)
2.397 ≈ 1.057^(t-50)
log(2.397) / log(1.057) ≈ t - 50
9.98 ≈ t - 50
t ≈ 60.98 or t ≈ 61°F
Therefore, the model predicts 1,000 visitors at a temperature of approximately 61°F.
The function f is invertible on its domain 50 < t ≤ 100 as it passes the horizontal line test. Each temperature value corresponds to a unique number of visitors.
To determine the predicted total amount of money paid by visitors when the daily high temperature is 78°F, we need to first find the number of visitors at 78°F using the function f:
f(78) = 416.939 * 1.057^(78-50) ≈ 956 visitors
Then, using the function g:
g(956) = 15 * 956 = $14,340
The function h is defined by h(t) = g(f(t)), meaning that the total amount of money paid by visitors is dependent on the number of visitors, which is in turn dependent on the daily high temperature.
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