Problem 16 - Applied Calculus

Question

Find a possible formula for the function represented by the data. $$ \begin{array}{c|c|c|c|c} \hline t & 0 & 1 & 2 & 3 \\ \hline g(t) & 5.50 & 4.40 & 3.52 & 2.82 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The function is \(g(t) = 5.50 \times 0.8^t\).

Step 1: Identify the Pattern

Look at the given values for \(t\) and \(g(t)\). Notice that as \(t\) increases, \(g(t)\) decreases. This suggests that \(g(t)\) might be an exponential decay function.

Step 2: Set Up the General Form

Assume \(g(t)\) follows the exponential decay form: \(g(t) = a \, b^t\), where \(a\) is the initial value and \(b\) is the growth factor (in this case, decay). We need to find \(a\) and \(b\).

Step 3: Determine the Initial Value

From the table, \(g(0) = 5.50\). Hence, the initial value \(a = 5.50\).

Step 4: Calculate the Decay Factor

Use the values at \(t=0\) and \(t=1\): \(g(0) = 5.50\) and \(g(1) = 4.40\). Substitute into \(g(1) = a\, b^1\) to get:\[4.40 = 5.50 \, b\]Solve for \(b\):\[b = \frac{4.40}{5.50} = 0.8\]

Step 5: Verify the Formula

Plug the values of \(a = 5.50\) and \(b = 0.8\) back into the equation and check the remaining values:\[g(t) = 5.50\, (0.8)^t\]For \(t = 2\), \(g(2) = 5.50 \, (0.8)^2 = 5.50 \, 0.64 = 3.52\). For \(t = 3\), \(g(3) = 5.50 \, (0.8)^3 = 5.50 \, 0.512 = 2.82\). The values match the table.

Key Concepts

Function Representation

In mathematics, understanding how to represent a function helps us uncover the underlying relationship between variables. Each entry in the provided table consists of an independent variable \( t \) and its corresponding dependent value \( g(t) \).
This table shows how \(g(t)\) changes as \(t\) increases.

• An exponential decay function is suggested by the pattern, as \(g(t)\) decreases over increasing values of \(t\).
• This type of function can often be expressed in the form \(g(t) = a \, b^t\), wherein the term \(a\) represents the initial quantity, and \(b\) is the decay factor.
• For the given data, \(g(t)\) decreases consistently and swiftly, an attribute of exponential functions.

To represent this accurately, we need to define both \(a\) and \(b\) through careful inspection of the values provided in the table. Identifying the pattern ensures the chosen function precisely describes the relationship between \(t\) and \(g(t)\).
This leads us to delve deeper into data analysis to confirm or adjust our assumptions.

Data Analysis

Data analysis involves examining the given dataset to discern the nature of relationships between variables. By analyzing columns \( t \) and \( g(t) \), we observe a steady decrease in \( g(t) \) as \( t \) increases.

• Our initial step is to recognize this trend as indicative of an exponential decay, where values fall by a consistent proportion.
• Through calculation, we identify the decay factor \( b \) by examining consecutive \( g(t) \) values. For instance, between \( g(0) = 5.50 \) and \( g(1) = 4.40 \), \( b \) computes as \( b = \frac{4.40}{5.50} \).
• The factor \( 0.8 \) indicates the proportion by which \( g(t) \) decreases per unit increase in \( t \).

This consistency affirms the assumption of an exponential relationship, solidifying the hypothesis that \( g(t) \) follows an exponential decay model. Data scrutinized here empowers us to establish accurate mathematical approximations and refine our conclusions.

Mathematical Modeling

Mathematical modeling is a powerful way to embody complex real-world situations through simplified mathematical representations.
This approach facilitates predictions and deeper insights into how systems behave over time.

• In this scenario, constructing a model to fit the data gives us \( g(t) = 5.50 \, (0.8)^t \).
• The model starts with an initial value \( a = 5.50 \), denoting the function's value at \( t = 0 \).
• The decay factor \( b = 0.8 \) represents how much \( g(t) \) decreases with each step increase in \( t \).

Verifying the model by plugging in various \( t \) values assures that the calculated \( g(t) \) aligns precisely with provided data values. In this way, mathematical modeling not only helps in understanding the current dataset but also allows us to predict future behaviors under similar conditions, emphasizing the encompassing nature of exponential decay.