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Ambient temperature refers to the temperature of the surrounding environment. It is a crucial factor in many scientific studies, as it can influence the behavior and characteristics of living organisms. In the context of the exercise, the ambient temperature is represented by the variable \( T \) and is measured in degrees Celsius. This variable plays a central role in determining the speed at which the ants run. The relationship is defined by the given formula \( S(T) = 0.2T - 2.7 \), indicating that as the ambient temperature changes, so does the speed of the ants. Understanding how temperature affects various processes is vital in fields such as biology, where temperature can influence growth rates and activity levels.

Functional notation provides a way to express relationships between variables using a function. It allows us to clearly state how one quantity depends on another. In this exercise, the speed \( S \) of the ants is expressed as a function of the ambient temperature \( T \), written as \( S(T) = 0.2T - 2.7 \). This notation means that if you know the temperature \( T \), you can calculate the speed \( S \) by substituting \( T \) into the function. For example, to find the speed at 30°C, you write \( S(30) \) and calculate it using the formula. Functional notation simplifies complex expressions and helps identify how variables are interconnected. It's a powerful tool in algebra that makes solving problems more efficient.

Rearranging formulas involves manipulating an equation to express one variable in terms of another. This is especially useful when you want to find the value of one variable based on the value of another. In the provided exercise, the original equation is \( S = 0.2T - 2.7 \). To find \( T \) in terms of \( S \), we rearrange the formula:

• Add 2.7 to both sides to get \( S + 2.7 = 0.2T \).
• Divide each side by 0.2, resulting in \( T = \frac{S + 2.7}{0.2} \).

This gives us a new formula that allows calculation of the ambient temperature if the speed is known. Rearranging equations helps delve deeper into the relationship between variables, and is a key skill in solving real-world problems.

Solving equations involves finding the value of unknowns that satisfy given mathematical statements. In this context, we are given a formula \( S = 0.2T - 2.7 \) where we know either the speed \( S \) or the temperature \( T \). When solving for a specific value, such as finding the ambient temperature when the speed is 3 cm/s, follow these steps:

• Use the rearranged formula \( T = \frac{S + 2.7}{0.2} \).
• Substitute the known value into the equation: \( T = \frac{3 + 2.7}{0.2} \).
• Calculate: first add 3 and 2.7 to get 5.7, then divide by 0.2 to find \( T = 28.5 \).

Solving equations is a fundamental skill in algebra that allows you to figure out unknowns in practical situations. Through practice, this process becomes a valuable tool for problem-solving across various applications.