Quantitative reasoning exercises using mathematical modeling

Quantitative reasoning exercises using mathematical modeling help develop analytical skills by applying mathematical concepts to solve real-world problems. Here are several exercises that incorporate different types of mathematical modeling, along with solutions or guiding steps.

Exercise 1: Linear Cost and Revenue Model

Problem: A company produces and sells widgets. The cost to produce $x$ widgets is given by the function $C(x) = 50x + 200$ (where $50 is the variable cost per widget and$200 is fixed costs). The selling price per widget is $80.

1. Write the revenue function $R(x)$.
2. Determine the number of widgets $x$ that must be sold to break even.

Solution Steps:

1. Revenue Function:

   $$R(x) = 80x$$

2. Break-even Point: Set the revenue equal to the cost:

   $$R(x) = C(x) \\ 80x = 50x + 200$$ $$30x = 200 \\ x = \frac{200}{30} \approx 6.67$$

   Since you can’t sell a fraction of a widget, the company must sell at least 7 widgets to break even.

Exercise 2: Exponential Growth Model

Problem: A bacteria culture starts with 200 bacteria and doubles every 3 hours.

1. Write a model for the population of bacteria $P(t)$ after $t$ hours.
2. How many bacteria will there be after 12 hours?

Solution Steps:

1. Population Model: Since the population doubles every 3 hours:

   $$P(t) = 200 \cdot 2^{\frac{t}{3}}$$

2. Population After 12 Hours:

   $$P(12) = 200 \cdot 2^{\frac{12}{3}} = 200 \cdot 2^4 = 200 \cdot 16 = 3200$$

   There will be 3,200 bacteria after 12 hours.

Exercise 3: Linear Programming for Optimization

Problem: A farmer has 100 acres of land to plant crops. Each acre of corn requires 2 hours of labor and yields $200 profit, while each acre of soybeans requires 4 hours of labor and yields$300 profit. The farmer has 240 hours of labor available.

1. Formulate the objective function for profit maximization.
2. Identify the constraints.
3. Determine how many acres of each crop to plant to maximize profit.

Solution Steps:

1. Variables: Let $x$ be the acres of corn and $y$ be the acres of soybeans.

2. Objective Function: Maximize profit $P$:

   $$P = 200x + 300y$$

3. Constraints:

   • Land constraint: $x + y \leq 100$
   • Labor constraint: $2x + 4y \leq 240$
   • Non-negativity: $x \geq 0, y \geq 0$

4. Solving the System: Use graphical methods or a linear programming tool to find the optimal solution.

Exercise 4: Population Decay Model

Problem: A certain radioactive substance has a half-life of 10 years. If you start with 80 grams of the substance, write an equation to model the amount remaining after $t$ years and calculate how much remains after 30 years.

Solution Steps:

1. Decay Model: The amount remaining after $t$ years can be modeled as:

   $$N(t) = N_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{10}}$$

   where $N_0 = 80$ grams.

2. Remaining After 30 Years:

   $$N(30) = 80 \cdot \left(\frac{1}{2}\right)^{\frac{30}{10}} = 80 \cdot \left(\frac{1}{2}\right)^3 = 80 \cdot \frac{1}{8} = 10$$

   After 30 years, 10 grams of the substance will remain.

Exercise 5: System of Equations in Economics

Problem: A coffee shop sells coffee for $5 per cup and pastries for$3 each. On a busy day, the shop sells 150 items for a total of $600.

1. Set up a system of equations to represent the situation.
2. Solve the system to find out how many cups of coffee and pastries were sold.

Solution Steps:

1. Variables: Let $c$ be the number of cups of coffee and $p$ be the number of pastries.

2. Equations:

   • Total items: $c + p = 150$ (1)
   • Total revenue: $5c + 3p = 600$ (2)

3. Solving the System: From equation (1), express $p$:

   $$p = 150 - c$$

   Substitute into equation (2):

   $$5c + 3(150 - c) = 600 \\ 5c + 450 - 3c = 600 \\ 2c = 150 \\ c = 75$$

   Then substitute back to find $p$:

   $$p = 150 - 75 = 75$$

   The shop sold 75 cups of coffee and 75 pastries.

Conclusion

These exercises demonstrate how mathematical modeling can be used to analyze various scenarios involving growth, decay, optimization, and systems of equations. By engaging with these problems, you can strengthen your quantitative reasoning skills and apply mathematical concepts to real-world situations.