Problem 45 - Basic Technical Mathematics with Calculus

Question

$$ \text { Solve the given quadratic equations by factoring.}$$ The power \(P\) (in MW) produced between midnight and noon by a nuclear power plant is \(P=4 h^{2}-48 h+744,\) where \(h\) is the hour of the day. At what time is the power 664 MW?

Step-by-Step Solution

Answer

The power is 664 MW at 2 AM and 10 AM.

Step 1: Identify the Given Equation

The problem states the power produced by a nuclear power plant is given by the quadratic equation \(P = 4h^2 - 48h + 744\). We need to determine when this power is 664 MW.

Step 2: Substitute Known Value

Replace \(P\) with 664 in the equation: \[4h^2 - 48h + 744 = 664\]

Step 3: Set the Equation to Zero

Subtract 664 from both sides to align the equation for factoring: \[4h^2 - 48h + 744 - 664 = 0\] which simplifies to \[4h^2 - 48h + 80 = 0\]

Step 4: Factor Out Common Term

Factor out the greatest common factor, which is 4: \[4(h^2 - 12h + 20) = 0\]

Step 5: Complete Factoring

Now factor \(h^2 - 12h + 20\). We need two numbers that multiply to 20 and add to -12: The numbers are -2 and -10. Rewrite as: \[4(h - 2)(h - 10) = 0\]

Step 6: Solve for Zero

Set each factor equal to zero: \(h - 2 = 0\) or \(h - 10 = 0\). This gives the solutions: \[h = 2\] or \[h = 10\]

Step 7: Interpret the Solutions

Since \(h\) represents the hour of the day, the solutions \(h = 2\) and \(h = 10\) correspond to 2 AM and 10 AM, respectively.

Key Concepts

Factoring Quadratics

To factor quadratics means to express a quadratic equation in its product form, typically looking something like \((x - p)(x - q) = 0\). By factoring, you can solve for the variable in a simple manner. Let's walk through it step by step:

• First, identify the quadratic equation. In our case, it was \(4h^2 - 48h + 80 = 0\).
• Factor out any greatest common factor, here it was 4: \(4(h^2 - 12h + 20) = 0\). This simplifies calculations.
• Find two numbers that multiply to give the constant term (20 in \(h^2 - 12h + 20\)) and add up to the coefficient of the linear term (-12). These numbers are -2 and -10.
• Express the quadratic as a product of two binomials: \((h - 2)(h - 10)\).

This factored form makes it easy to find solutions. Just set each factor to zero to solve for the variable. Factoring is an essential skill, especially in more complex mathematical problems.

Nuclear Power Calculations

Nuclear power plants generate electricity through nuclear reactions. Calculating power output is crucial for operations and safety. The function we analyzed, \(P=4h^{2}-48h+744\), gives the power output in megawatts at any hour, \(h\):

• The quadratic part, \(4h^2\), reflects changes in output over the hours.
• The linear component, \(-48h\), adjusts the curve, reflecting variables like fuel use and thermal efficiency.
• The constant, \(744\), is the base level of power when all other variables are balanced.

Understanding these calculations allows engineers to adjust operations to ensure optimal power delivery. They help predict peak outputs and plan maintenance schedules, making them critical for efficient plant operations.

Real-World Applications of Mathematics

Mathematics is everywhere in real life. Quadratic equations, like the one used in this problem, often arise in daily life. In our task, it applied to calculating power output for a nuclear plant:

• Engineering: Engineers apply quadratic equations to design structures and solve mechanics problems.
• Economics: Analysts use them to calculate profit maximization and cost predictions.
• Physics: Motion under gravity problems and projectile motion are classic quadratic cases.

By seeing mathematics in real-world contexts, students understand its importance and its practical applications. From energy efficiency in plants to strategic business decisions, math strengthens problem-solving capabilities.

Solving Equations

Solving equations is fundamental in math. It involves finding values that satisfy given conditions or formulas. In our exercise, solving \(4(h - 2)(h - 10) = 0\) involved:

• Setting each factor equal to zero: \(h - 2 = 0\) and \(h - 10 = 0\).
• Calculating the solutions: \(h = 2\) and \(h = 10\).
• Interpreting these solutions in context: They represent 2 AM and 10 AM, the times when the power output is 664 MW.

Solving equations is a building block for more advanced math topics. It allows students to develop logical thinking and an analytical approach, which is crucial in tackling real-world challenges. By practicing solving various mathematical equations, one refines these essential skills.