Problem 80 - A Graphical Approach to College Algebra

Question

Solve each problem. Recycling \(\quad\) A cost-benefit function \(C\) computes the cost in millions of dollars of implementing a city recycling project when \(x\) percent of the citizens participate, where $$C(x)=\frac{1.2 x}{100-x}$$ (a) Graph \(C\) in the window \([0,100]\) by \([0,10] .\) Interpret the graph as \(x\) approaches \(100 .\) (b) If \(75 \%\) participation is expected, determine the cost for the city. (c) The city plans to spend \(\$ 5\) million on this recycling project. Estimate graphically the percentage of participation that they are expecting. (d) Solve part (c) analytically.

Step-by-Step Solution

Answer

75% participation costs $3.6M; $5M cost implies ~80.65% participation analytically.

Step 1: Analyze the Given Function

The given cost-benefit function is \( C(x) = \frac{1.2x}{100-x} \). This function describes how the cost of implementing a recycling project depends on the participation percentage of the citizens \( x \). The function is undefined at \( x = 100 \) since the denominator becomes zero, indicating a vertical asymptote at \( x = 100 \).

Step 2: Graph the Function

Using a graphing tool, plot the function \( C(x) = \frac{1.2x}{100-x} \) in the window \([0,100]\) for \( x \) and \([0,10]\) for \( y \). As \( x \) approaches \( 100 \), observe that \( C(x) \) rises sharply towards infinity, showing that costs become unmanageable as citizen participation nears 100%.

Step 3: Calculate Cost for 75% Participation

Given \( x = 75 \%, \) substitute into the formula: \[ C(75) = \frac{1.2 \times 75}{100 - 75} = \frac{90}{25} = 3.6 \]Thus, the cost is \( 3.6 \) million dollars.

Step 4: Estimate Participation for $5 Million

Graphically, locate the point on the graph where \( C(x) = 5 \). Draw a horizontal line at \( y = 5 \) and find where it intersects the curve. Estimate the \( x \)-value at this intersection; it is around \( 85 \% \).

Step 5: Solve for Participation Analytically

To solve \( C(x) = 5 \), set up the equation: \[ \frac{1.2x}{100-x} = 5 \]Cross-multiply to obtain: \[ 1.2x = 5(100-x) \]Expand and solve for \( x \):\[ 1.2x = 500 - 5x \6.2x = 500 \x = \frac{500}{6.2} \approx 80.65 \% \] Thus, the participation is approximately \( 80.65 \% \).

Key Concepts

Recycling Economics

Recycling economics involves understanding the balance between costs and benefits associated with waste management projects, like recycling. This balance is crucial for making informed decisions about implementing city-wide initiatives. In the given problem, the cost-benefit function \(C(x) = \frac{1.2x}{100-x}\) is used to model how the cost in millions of dollars changes with citizen participation, \(x\). The idea here is to weigh the financial investment against the environmental and communal benefits of recycling. As more residents engage in recycling, the cost tends to escalate, especially as participation nears totality. Understanding this relationship aids in budgeting and planning effective recycling programs that maximize participation at manageable costs.

Graphical Analysis

Graphical analysis helps visualize mathematical functions to better understand the relationship between variables. For the function \(C(x) = \frac{1.2x}{100-x}\), graphing it over the intervals \([0,100]\) for the \(x\)-axis and \([0,10]\) for the \(y\)-axis illustrates how the cost behavior changes as participation increases.
As \(x\) approaches 100%, the cost graph steepens drastically and approaches infinity due to the vertical asymptote at \(x = 100\). This visual representation provides valuable insight into how costs increase disproportionately as participation nears 100%, making it clear that encouraging full participation might be economically impractical. Graphs make complex numerical relationships more intuitive, aiding in decision-making processes for municipal administrations.

Participation Rate

The participation rate in a recycling program is a key determinant of the overall cost, as modeled by the function \(C(x)\). In practical scenarios, knowing the anticipated participation rate helps city planners allocate funds accurately. For instance, if a 75% participation rate is expected, the cost is calculated by substituting into the function:
\[ C(75) = \frac{1.2 \times 75}{100 - 75} = 3.6 \]
Million dollars. This demonstrates how a specific participation target influences the budget.

• The participation rate affects policy decisions.
• Projects can aim for feasible rates that align with financial constraints.

Analyzing and predicting participation rates allows for better financial planning, ensuring recycling goals are economical and effective.

Vertical Asymptote in Functions

A vertical asymptote in a function like \(C(x) = \frac{1.2x}{100-x}\) occurs where the function is undefined, at \(x = 100\). This line indicates that the cost tends toward infinity as \(x\) nears 100%. Vertical asymptotes represent limits and restrictions within a mathematical model, highlighting conditions under which the model breaks down.
In the context of the exercise, the vertical asymptote illustrates that while 100% participation is theoretically possible, the cost of achieving this becomes unmanageable in practice. This informs planners that seeking complete participation may not be feasible and that optimal points before reaching the asymptote should be identified for sustainable project implementation. Understanding function behavior around vertical asymptotes helps in designing realistic and efficient policies.