Problem 153 - Elementary Algebra

Question

A farmer plans to fence off sections of a rectangular corral. The diagonal distance from one corner of the corral to the opposite corner is five yards longer than the width of the corral. The length of the corral is three times the width. Find the length of the diagonal of the corral. Round to the nearest tenth.

Step-by-Step Solution

Answer

The length of the diagonal is 7.2 yards.

Step 1: Define Variables

Let the width of the corral be denoted as \( w \). According to the problem, the length \( l \) of the corral is three times the width, so \( l = 3w \).

Step 2: Set Up Relationship for the Diagonal

The problem states that the diagonal distance is five yards longer than the width. Therefore, if the diagonal is denoted by \( d \), we have \( d = w + 5 \).

Step 3: Apply the Pythagorean Theorem

For a rectangle, the diagonal can be determined using the Pythagorean Theorem: \[ d^2 = l^2 + w^2 \]. Substitute the values for \( l \) and \( d \) into this equation.

Step 4: Substitute Values and Simplify

Substitute \( l = 3w \) and \( d = w + 5 \) into the Pythagorean Theorem: \[ (w + 5)^2 = (3w)^2 + w^2 \]. Simplify the equation: \[ (w + 5)^2 = 9w^2 + w^2 \] or \[ w^2 + 10w + 25 = 10w^2 \].

Step 5: Rearrange to Form a Quadratic Equation

Rearrange the equation to form a quadratic equation: \[ 10w^2 - w^2 - 10w - 25 = 0 \]. Simplify to get: \[ 9w^2 - 10w - 25 = 0 \].

Step 6: Solve the Quadratic Equation

Solve the quadratic equation \( 9w^2 - 10w - 25 = 0 \) using the quadratic formula \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 9 \), \( b = -10 \), and \( c = -25 \).

Step 7: Calculate the Width

Plug in the values: \[ w = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 9 \cdot (-25)}}{2 \cdot 9} \] \[ w = \frac{10 \pm \sqrt{100 + 900}}{18} \] \[ w = \frac{10 \pm 30}{18} \]. So \( w = \frac{40}{18} = 2.22 \) or \( w = \frac{-20}{18} = -1.11 \). Only positive value is acceptable, so \( w = 2.22 \).

Step 8: Find the Diagonal

Substitute \( w \) back to find \( d \): \[ d = w + 5 \] \[ d = 2.22 + 5 \] \[ d = 7.2 \].

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry. It applies to right-angled triangles and helps you find the length of one side when you know the lengths of the other two. The theorem states that for any right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, this is expressed as:








For the corral problem, we used the Pythagorean Theorem to relate the diagonal (hypotenuse) with the length and width of the rectangle (legs). Recognizing the corral as a right-angled triangle placed diagonally helped simplify the problem.

Solving Quadratic Equations

Quadratic equations are equations of the form

Quadratic equations are usually solved using the quadratic formula:
This formula helps you find the values of the variable that satisfy the equation. It's especially useful when equations can't be factored easily.
For the corral's problem, we re-arranged the equation to obtain a standard quadratic equation, and then used the quadratic formula to find the width of the corral.

Algebraic Problem-Solving

Algebraic problem-solving involves forming equations based on given conditions and simplifying them to find the solution. This may include:

• Defining variables based on given relationships
• Translating word problems into mathematical equations
• Solving the equations to find the unknowns
• Rechecking all steps and calculations
• Applying learned formulas such as the Pythagorean theorem or quadratic formula

In this exercise, we set variables for the width, length, and diagonal of the corral. By following a series of logical algebraic steps, we derived a quadratic equation and solved it to find the width and diagonal.

Geometric Relationships

Understanding geometric relationships is key to solving many real-world problems. Knowing how angles, lengths, and shapes interact allows us to build equations and sets of relationships.
For instance, in a rectangle:

• Opposite sides are equal in length
• The diagonal forms two congruent right-angled triangles
• The Pythagorean theorem can be employed on these triangles to link the sides and diagonal
• Specific given conditions (like the diagonal being longer than one side) provide additional relations to form equations

These relationships helped us relate the rectangular corral's diagonal to its length and width, eventually allowing us to solve the problem step-by-step.