Problem 17 - Algebra Form and Function

Question

Identify the initial value and the rate of change, and explain their meanings in practical terms. On a spring day the temperature in degrees Fahrenheit is \(50+1.2 h,\) where \(h\) is the number of hours since noon.

Step-by-Step Solution

Answer

Answer: The initial value is 50 degrees Fahrenheit, which represents the temperature at noon. The rate of change is 1.2, which means that for every hour since noon, the temperature increases by 1.2 degrees Fahrenheit.

Step 1: Find the Initial Value

To find the initial value, we need to evaluate the function at \(h=0\). Plugging in \(h=0\), we get: \(50+1.2(0)=50\) The initial value is \(50\). This means that the temperature at noon is \(50\) degrees Fahrenheit.

Step 2: Find the Rate of Change

The rate of change is the coefficient of \(h\) in the given function, which is \(1.2\). This means that as \(h\) increases by \(1\), the temperature increases by \(1.2\) degrees. In practical terms, the temperature rises by \(1.2\) degrees Fahrenheit for every hour that passes since noon.

Key Concepts

Understanding the Initial Value

When analyzing a linear function like the one given for the temperature change, the initial value is a crucial concept. It represents the value of the function at the starting point, which occurs when the independent variable is zero. In this example, the function is expressed as:\[ f(h) = 50 + 1.2h \]To identify the initial value, you set \( h = 0 \). When calculated, this yields:\[ 50 + 1.2(0) = 50 \]Thus, the initial value is 50. What this tells us is that the temperature at noon, before any time has passed, is 50 degrees Fahrenheit. The initial value in the formula is often signified as the constant term, and it provides pivotal information about where the function begins.

Decoding the Rate of Change

The rate of change in our temperature function is found by looking at the coefficient of \( h \), which is 1.2. In a linear function, the rate of change tells us how much the dependent variable changes with respect to a change in the independent variable.For the function \( f(h) = 50 + 1.2h \):

• The coefficient \( 1.2 \) is the rate of change.
• This implies that for each hour past noon (i.e., for every \( h = 1 \)), the temperature increases by 1.2 degrees Fahrenheit.

In simple terms, it describes the speed at which the temperature is rising. A positive rate, like 1.2, means the temperature is increasing, which in this context, accounts for the warming up as the afternoon progresses on a pleasant spring day.

Practical Applications in Daily Life

The concepts of initial value and rate of change have practical applications in everyday life. Think about planning your day based on expected temperature changes. If you know the initial temperature and how quickly it's changing, you can decide the best time for activities. For example:

• If you're planning an outdoor event and see that the temperature starts at 50 degrees and increases by 1.2 degrees per hour, you can estimate when it will be most comfortable.
• In business, understanding these concepts can help in financial forecasting, such as predicting sales growth over time.

These insights enable better decision-making based on expected future conditions, which helps manage plans and resources efficiently.