Solving Word Problems Involving Quadratics

Quadratic equations can often be used to solve real-world problems that involve relationships between quantities. By setting up an equation based on the information given, you can solve for unknowns and answer questions about the scenario.

Steps for Solving Quadratic Word Problems

  1. Understand the problem: Read the problem carefully to identify what is being asked and the information provided.
  2. Define the variables: Assign variables to unknown quantities, and set up an equation based on the relationships described in the problem.
  3. Formulate a quadratic equation: Write down an equation in the form \( ax^2 + bx + c = 0 \).
  4. Solve the equation: Use the quadratic formula, factoring, or completing the square to find the values of the unknown variable(s).
  5. Interpret the solution: Check if your answer makes sense within the context of the problem.

Example Problem

A farmer has 200 meters of fencing and wants to create a rectangular enclosure along a river, so only three sides need fencing. If the length along the river is \( l \) and the width is \( w \), find the dimensions that maximize the area.

Solution:

  • Step 1: Define \( w \) as the width and \( l = 200 - 2w \) as the length.
  • Step 2: The area \( A \) is \( w \times (200 - 2w) \), so \( A = 200w - 2w^2 \).
  • Step 3: Rewrite as \( A = -2w^2 + 200w \), a quadratic equation.
  • Step 4: Find the maximum area by locating the vertex. Here, \( w = 50 \) meters gives the maximum area.
  • Step 5: Substitute \( w = 50 \) to find \( l = 100 \). The dimensions that maximize area are 50 meters by 100 meters.

Practice Questions

  1. A rectangular garden has a length that is 4 meters more than twice its width. If the area of the garden is 96 square meters, what are the dimensions of the garden?
    Solution

    Let \( w \) be the width. Then, the length \( l = 2w + 4 \).

    Form the equation: \( w(2w + 4) = 96 \), simplifying to \( 2w^2 + 4w - 96 = 0 \).

    Solving this quadratic gives \( w = 6 \) meters and \( l = 16 \) meters.

    The dimensions are 6 meters by 16 meters.

  2. A ball is thrown into the air with an initial upward velocity of 30 m/s from a height of 2 meters. The height \( h \) (in meters) of the ball after \( t \) seconds is given by the equation:

    $$ h = -5t^2 + 30t + 2 $$

    When will the ball hit the ground?

    Solution

    Set \( h = 0 \) and solve: \( -5t^2 + 30t + 2 = 0 \).

    Using the quadratic formula, \( t \approx 6.1 \) seconds.

    The ball will hit the ground after approximately 6.1 seconds.

  3. A company finds that the cost \( C \) (in dollars) to produce \( x \) units of a product is given by:

    $$ C = x^2 - 20x + 300 $$

    How many units should the company produce to minimize the cost?

    Solution

    Minimize \( C \) by finding the vertex of the quadratic function. The vertex formula gives \( x = 10 \).

    The minimum cost occurs when the company produces 10 units.

→Introduction