Question

State a possible domain and range for the given functions, which represent real-world applications. a) The population of a lakeside town with large numbers of seasonal residents is modelled by the function \(P(t)=6000 \sin (t-8)+8000.\) b) The height of the tide on a given day can be modelled using the function \(h(t)=6 \sin (t-5)+7\) c) The height above the ground of a rider on a Ferris wheel can be modelled by \(h(t)=6 \sin 3(t-30)+12.\) d) The average daily temperature may be modelled by the function \(h(t)=9 \cos \frac{2 \pi}{365}(t-200)+14.\)

Short answer

(a) Domain: all real numbers, Range: [2000, 14000]. (b) Domain: all real numbers, Range: [1, 13]. (c) Domain: all real numbers, Range: [6, 18]. (d) Domain: all real numbers, Range: [5, 23].

Step-by-step solution

- Analyze the Population Function

The function given is \(P(t) = 6000 \sin(t-8) + 8000\). Since \( \sin(t) \) oscillates between -1 and 1, the minimum value of \( \sin(t) \) is -6000 and the maximum value is 6000. Adding 8000 shifts the range: \[ \min P(t) = 8000 - 6000 = 2000 \]\[ \max P(t) = 8000 + 6000 = 14000 \]Thus, the range of \(P(t)\) is from 2000 to 14000. There is no restriction stated on \(t\), so the domain is all real numbers.

- Analyze the Tide Height Function

The function given is \(h(t) = 6 \sin(t-5) + 7\). Since \( \sin(t) \) oscillates between -1 and 1, the minimum value of \( \sin(t) \) is -6 and the maximum value is 6. Adding 7 shifts the range: \[ \min h(t) = 7 - 6 = 1 \]\[ \max h(t) = 7 + 6 = 13 \]Thus, the range of \(h(t)\) is from 1 to 13. There is no restriction stated on \(t\), so the domain is all real numbers.

- Analyze the Ferris Wheel Function

The function given is \(h(t) = 6 \sin 3(t-30) + 12\). Since \( \sin(t) \) oscillates between -1 and 1, the minimum value of \( \sin(t) \) is -6 and the maximum value is 6. Adding 12 shifts the range: \[ \min h(t) = 12 - 6 = 6 \]\[ \max h(t) = 12 + 6 = 18 \]Thus, the range of \(h(t)\) is from 6 to 18. There is no restriction stated on \(t\), so the domain is all real numbers.

- Analyze the Temperature Function

The function given is \(h(t) = 9 \cos( \frac{2\pi}{365} (t-200) ) + 14\). Since \( \cos(t) \) oscillates between -1 and 1, the minimum value of \( \cos(t) \) is -9 and the maximum value is 9. Adding 14 shifts the range: \[ \min h(t) = 14 - 9 = 5 \]\[ \max h(t) = 14 + 9 = 23 \]Thus, the range of \(h(t)\) is from 5 to 23. There is no restriction stated on \(t\), so the domain is all real numbers.

Consolidate Results

Summarizing the range and domain of each function: (a) \(P(t)\) has domain all real numbers and range 2000 to 14000.(b) \(h(t)\) has domain all real numbers and range 1 to 13.(c) \(h(t)\) has domain all real numbers and range 6 to 18.(d) \(h(t)\) has domain all real numbers and range 5 to 23.

Key concepts

Population Model

A population model is a mathematical function that represents how a population changes over time. In this case, the population of a lakeside town with many seasonal residents is modeled by the function: \[ P(t) = 6000 \, \text{sin}(t-8) + 8000 \]Here, 'P(t)' represents the population at time 't'. The sine function, \(\text{sin}(t-8)\), oscillates between -1 and 1, creating a cyclical pattern which is ideal for modeling seasonal variations. The multiplication by 6000 scales this oscillation, showing how much the population can increase or decrease. The addition of 8000 represents a baseline population, shifting the whole function upwards. This makes the minimum population 2000 (when \(\text{sin}(t-8)\) is -1) and the maximum 14000 (when \(\text{sin}(t-8)\) is 1). The domain, representing all possible times 't', is all real numbers, and the range is from 2000 to 14000.

Sine Function

The sine function, denoted as \( \text{sin}(t) \), is a periodic function that oscillates between -1 and 1. This continuous wave-like pattern makes it perfect for modeling cyclical phenomena in the real world. In the context of the given problems:

• \[ h(t) = 6 \, \text{sin}(t-5) + 7 \] models tide heights.
• \[ h(t) = 6 \, \text{sin}(3(t-30)) + 12 \] models the height of a Ferris wheel ride.

These functions use different scaling and shifting constants to adapt the basic sine wave to match real-world data. In general, a sine function \( A \, \text{sin}(B(t-C)) + D \) consists of:

• A - Amplitude: Determines the height of the peaks (e.g., 6000 in the population model).
• B - Frequency: Determines the number of cycles within a given period (e.g., 3 in the Ferris wheel model).
• C - Phase shift: Moves the wave horizontally (e.g., -8 in the population model).
• D - Vertical shift: Moves the wave up or down to adjust the baseline (e.g., 8000 in the population model).

Cosine Function

The cosine function, denoted as \( \text{cos}(t) \), is also a periodic function similar to the sine function but starts at its maximum value. It oscillates between -1 and 1 just like the sine function. In the context of the given problems, the cosine function models the average daily temperature: \[ h(t) = 9 \, \text{cos} \left( \frac{2\pi}{365}(t-200) \right) + 14 \]This temperature model oscillates annually (with a period of 365 days), reaching a minimum value of 5 and a maximum of 23 degrees. The components of the function are:

• Amplitude (9) - Determines the temperature variation.
• Frequency (2π/365) - Sets the annual cycle.
• Phase shift (-200) - Aligns the model to seasons and specific days.
• Vertical shift (14) - Adjusts the baseline average temperature.

The cosine function's applicability in modeling temperature helps us understand seasonal weather changes effectively.

Real-World Applications

Mathematical functions like sine and cosine have numerous real-world applications due to their periodic nature. They help us model and understand scenarios that naturally repeat over time. Some examples from the exercise are:

• Population models to predict seasonal changes in towns.
• Tide height prediction to assist in marine and coastal activities.
• Ferris wheel motion to entertain and ensure rider safety.
• Temperature variations to plan for agricultural and climatic studies.

These models facilitate predictions, planning, and optimization in various fields. For instance, understanding the cyclic nature of populations can aid in resource allocation, while tide models are crucial for navigation and coastal management. The versatility of these functions makes them indispensable tools in scientific, engineering, and economic planning processes.