Problem 55 - Algebra for College Students

Question

Rod agreed to mow a vacant lot for $$\$ 12$$. It took him an hour longer than he had anticipated, so he earned $$\$ 1$$ per hour less than he had originally calculated. How long had he anticipated that it would take him to mow the lot?

Step-by-Step Solution

Answer

Rod had anticipated it would take him 3 hours to mow the lot.

Step 1: Understanding the Variables Involved

Let \( t \) be the time (in hours) Rod had anticipated it would take to mow the lot. Therefore, his anticipated hourly rate would be \( \frac{12}{t} \) dollars per hour.

Step 2: Setting Up the Equation

Since it took him an hour longer than anticipated, the actual time taken is \( t+1 \). Thus, his actual hourly rate is \( \frac{12}{t+1} \) dollars per hour. We know from the problem that he earned \$1 less per hour than anticipated: \( \frac{12}{t} - \frac{12}{t+1} = 1 \).

Step 3: Simplifying the Equation

Simplify the equation: \[ \frac{12}{t} - \frac{12}{t+1} = 1 \]Multiply every term by \( t(t+1) \) to eliminate the fractions:\[ 12(t+1) - 12t = t(t+1) \]

Step 4: Solving the Simplified Equation

Distribute and rearrange the equation:\[ 12t + 12 - 12t = t^2 + t \]This simplifies to:\[ t^2 + t - 12 = 0 \]

Step 5: Factoring the Quadratic Equation

Factor the quadratic equation:\[ (t+4)(t-3) = 0 \]This gives possible solutions for \( t \) as \( t = -4 \) or \( t = 3 \). As time cannot be negative, \( t = 3 \).

Step 6: Verifying the Solution

Rod had anticipated it would take him 3 hours. If he actually took 4 hours, his rate was \\(3 per hour, instead of \\)4 per hour as originally calculated, which means he earned \$1 less per hour, matching the conditions of the problem.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a eq 0\). The goal is to find the value(s) of \(x\) that satisfy the equation. These equations can help solve complex real-life situations like our mowing example.

In the given problem, Rod needs to solve the quadratic equation \(t^2 + t - 12 = 0\). This equation was derived from setting up the problem about his anticipated and actual working time. Quadratic equations are often solved using different methods such as:

• Factoring: Breaking down the equation into two binomial expressions.
• Completing the Square: Rewriting the equation to make it easier to solve.
• Quadratic Formula: Using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) for direct solutions.

Rod's problem was ideal for solving by factoring, thanks to the factors \((t+4)(t-3)\). Hence, the solution concludes with plausible values for \(t\) being 3 hours.

Problem Solving

Problem-solving involves identifying the elements of a problem, forming equations, and finding a solution. Here, Rod's issue was managing time and pay for mowing a lawn, a common real-world problem.

To solve problems, consider these structured steps:

• Define the Variables: Identify unknowns. Here it's the time Rod anticipated, represented by \(t\).
• Develop the Equation: Translate the word problem into a mathematical expression (\(\frac{12}{t} - \frac{12}{t+1} = 1\)).
• Simplify: Clear fractional terms and move towards a quadratic equation.
• Analyze Results: Evaluate all potential solutions. Eliminate non-feasible ones (e.g., negative time).

By systematically breaking down the problem, we effectively apply mathematical principles to find that Rod initially expected to work for 3 hours.

Linear Equations

Linear equations involve variables raised to the power of one. They are straightforward and foundational in understanding more complex equations like quadratics. While not explicitly the final step in Rod's solution, understanding the linear part helps break down how changes in time affected his hourly rate.

At the heart of the initial problem was a linear concept—the relationship between time and hourly rate:

• The equation \(\frac{12}{t} - \frac{12}{t+1} = 1\) involves linear operations that are crucial before forming the quadratic.
• Understanding this simplifies the relationship between time and wage, highlighting the basic idea that a longer time worked at a lower hourly rate.

Grasping linear equations helps view the transition from the situation's setup to the quadratic form, forming a baseline for solving through a structured algebraic approach.