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.
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.
1 answer