Problem 30 - Algebra Form and Function

Question

Give the domain and range of the functions described. Let \(N=f(H)\) given the number of days it takes a certain kind of insect to develop as a function of the temperature \(H\left(\right.\) in \(\left.{ }^{\circ} \mathrm{C}\right)\). At \(40^{\circ} \mathrm{C}-\) the maximum it can tolerate- the insect requires 10 full days to develop. An additional day is required for every \(2^{\circ} \mathrm{C}\) drop, and it cannot develop in temperatures below \(10^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

Answer: The domain of the function is \(10\leq H\leq 40\), and the range of the function is \(10\leq N\leq 25\).

Step 1: Identify the function's constraints

The function's constraints are given as follows: - The maximum temperature the insect can tolerate is \(40^{\circ}\)C. - The number of days required for the insect to develop at this maximum temperature is 10 days. - The insect requires one additional day for every \(2^{\circ}\)C drop in temperature. - The insect cannot develop in temperatures below \(10^{\circ}\)C.

Step 2: Determine the domain

The domain of the function represents the possible temperature values (H). From the given constraints, we can observe that the minimum temperature is \(10^{\circ}\)C, and the maximum temperature is \(40^{\circ}\)C. Therefore, the domain of the function is \(10\leq H\leq 40\).

Step 3: Determine the range

The range of the function represents the possible number of days for the insect to develop (N). The insect takes 10 days to develop at the maximum temperature of \(40^{\circ}\)C. When the temperature decreases by \(2^{\circ}\)C, the insect requires one more day. The process continues until the minimum temperature of \(10^{\circ}\)C is reached. To find the maximum number of days required, we need to calculate the temperature difference and see how many additional days are needed: Temperature difference = \(40^{\circ}\)C - \(10^{\circ}\)C = \(30^{\circ}\)C Number of additional days = \(30^{\circ}\)C / \(2^{\circ}\)C = 15 days Therefore, at the minimum temperature of \(10^{\circ}\)C, the insect takes 10 + 15 = 25 days to develop. Hence, the range of the function is \(10\leq N\leq 25\).

Key Concepts

Functions

In mathematics, a function is a special relationship where each input has a single output. You can think of it like a machine: you put something in, and the machine gives you something back based on a specific rule. In the exercise, the function is defined as \( N=f(H) \), which tells us that the number of days \( N \) it takes for an insect to develop depends on the temperature \( H \). This is a typical example of a function, where the temperature is our input, and the number of days is our output.

• Inputs and Outputs: The input \((H)\) is the temperature at which the insect is developing.
• Function Rule: Every \(2^{\circ}\text{C}\) drop from top temperature adds one extra development day.

The concept of a function helps us model real-life scenarios mathematically. It simplifies the understanding of how changes in temperature affect the insect's development time. Understanding functions is crucial in mathematics as they form the basis of automation and algorithms in computing, graph plotting, and much more.

Temperature

Temperature plays a significant role in many biological processes and is crucial in understanding this insect development scenario. The temperature can be viewed as a scale that determines how quickly or slowly specific biological functions occur.
In our problem, the temperature restriction is from \(10^{\circ}\text{C}\) to \(40^{\circ}\text{C}\). This vital range is crucial for the insect's development.

• Maximum Limit: At \(40^{\circ}\text{C}\), the insect develops in the shortest time, which is 10 days.
• Minimum Limit: Below \(10^{\circ}\text{C}\), the insect cannot develop, establishing the lower temperature bound.

Temperature fluctuations directly affect the insect's life cycle. As temperatures drop, insects need more time to develop because lower temperatures slow their metabolic rates. Hence, understanding temperature dynamics helps in predicting insect behavior and planning in agriculture or environmental sciences.

Constraints

Constraints in functions define the limits within which the inputs and outputs can exist. These play a critical role in ensuring that functions remain realistic and applicable to real-world phenomena.
In our scenario, the constraints are set by the biological limits of the insect's development:

• Upper Temperature Limit: The insect can develop up to a maximum temperature of \(40^{\circ}\text{C}\).
• Lower Temperature Limit: Below \(10^{\circ}\text{C}\), it is incapable of development.

Constraints help us understand what values are sustainable and meaningful in a given situation. They ensure the mathematical model reflects reality closely and governs the domain and range of the function: ensuring \(10 \leq H \leq 40\) for the domain and \(10 \leq N \leq 25\) for the range. This application of constraints is crucial in decision-making and strategic planning across various fields, such as engineering, science, and business, by defining what scenarios are feasible.