Problem 55 - College Algebra

Question

For the following exercises, refer to Table 12. $$\begin{array}{ccccccccccc}{x} & {0} & {2} & {4} & {5} & {7} & {8} & {10} & {11} & {15} & {17} \\ {f(x)} & {12} & {28.6} & {52.8} & {70.3} & {99.9} & {112.5} & {125.8} & {127.9} & {135.1} & {135.9}\end{array}$$ Use the intersect feature to find the value of x for which the model reaches half its carrying capacity.

Step-by-Step Solution

Answer

The value of \(x\) is approximately 4.87 when the model reaches half its carrying capacity.

Step 1: Identify the model's carrying capacity

The carrying capacity is the maximum value that the function \(f(x)\) approaches. From the table, the highest \(f(x)\) value is 135.9. Hence, the carrying capacity is \(135.9\).

Step 2: Calculate half the carrying capacity

To find half the carrying capacity, divide the carrying capacity by 2. \[ \frac{135.9}{2} = 67.95 \] So, half the carrying capacity is \(67.95\).

Step 3: Locate the x-value for half carrying capacity

Identify the \(x\) value where \(f(x)\) is closest to \(67.95\). From the table, the closest \(f(x)\) values around 67.95 are 52.8 (at \(x=4\)) and 70.3 (at \(x=5\)).

Step 4: Use interpolation for precise intersect

To find a more precise \(x\) value, use linear interpolation between the points (4, 52.8) and (5, 70.3). Calculate the interpolated \(x\) for \(f(x) \approx 67.95\).The formula for linear interpolation is:\[ x = x_1 + \frac{(y - y_1)(x_2 - x_1)}{y_2 - y_1} \]Substitute \(x_1 = 4\), \(y_1 = 52.8\), \(x_2 = 5\), \(y_2 = 70.3\), and \(y = 67.95\):\[ x = 4 + \frac{(67.95 - 52.8)(5 - 4)}{70.3 - 52.8} \]\[ x \approx 4 + \frac{15.15}{17.5} \]\[ x \approx 4.87 \]

Step 5: Conclude the result

The value of \(x\) when \(f(x)\) is half the carrying capacity (67.95) is approximately \(4.87\).

Key Concepts

Carrying Capacity

Carrying capacity is a term often used in both biology and mathematics. It represents the maximum value or limit that a system, organism, or model can reach. In mathematical models, particularly those depicting growth, the carrying capacity is the plateau point where the growth stabilizes.
In the context of the table from this exercise, carrying capacity is the highest function value, which, based on the table data, is 135.9. This means that the function approaches a maximum or steady state of 135.9.
Understanding carrying capacity helps in determining the potential limits of a system and is a key concept in modeling scenarios such as population growth and logistic growth. When analyzing data, identifying the carrying capacity can provide insights into the behavior and limitations of the model.

Linear Interpolation

Linear interpolation is a useful mathematical technique used to estimate or find values between two known values on a line or curve. This is often used when precise value measurements are unavailable, and estimation is necessary.
To perform linear interpolation, you calculate where a specific value falls between two points. The interpolation formula used here is:- Identify the known points: - For example, in the exercise, points are (4, 52.8) and (5, 70.3).- Use the linear interpolation formula:\[x = x_1 + \frac{(y - y_1)(x_2 - x_1)}{y_2 - y_1}\]- Substitute the given and desired function values.Linear interpolation is especially handy for functions that don't provide explicit outputs for all input values. It helps you to find an approximation without additional complex calculations or needing a specific function expression.

Function Values

Function values refer to the outputs of a function for specific input values. They are typically represented as \(f(x)\), where \(x\) is the input, and \(f(x)\) is the corresponding output.
In this exercise's table, each \(x\) has an associated \(f(x)\), representing the relationship between inputs and outputs of the given model.
Analyzing function values allows for understanding how a function behaves across different inputs. It can be particularly important for identifying trends, calculating interpolations, or determining intersections. By examining these values, we can derive insights into the mathematical model, such as locating the input value that results in half the carrying capacity, enhancing comprehension of the model's dynamics. Function values are core to understanding any relationship described by a function.