Problem 33 - Beginning and Intermediate Algebra

Question

Write an equation and solve. A 13 -ft ladder is leaning against a wall. The distance from the top of the ladder to the bottom of the wall is \(7 \mathrm{ft}\) more than the distance from the bottom of the ladder to the wall. Find the distance from the bottom of the ladder to the wall.

Step-by-Step Solution

Answer

Using the Pythagorean theorem and given information, we form the equations \(x^2+y^2=13^2\) and \(y=x+7\). Substituting and solving for x, we find the valid solution is \(x=5\), meaning the distance from the bottom of the ladder to the wall is 5 ft.

Step 1: Write the equation using the Pythagorean theorem.

The ladder forms a right-angled triangle with one leg as x (distance from the bottom of the ladder to the wall) and another leg as y (distance from the top of the ladder to the bottom of the wall). Using the Pythagorean theorem, we have:\[x^2 + y^2 = 13^2\]

Step 2: Use given information to form a second equation.

Since the distance from the top of the ladder to the bottom of the wall is 7 ft more than the distance from the bottom of the ladder to the wall, we form the second equation: \[y = x + 7\]

Step 3: Substitute the second equation into the first equation.

Now, we will substitute y from the second equation into the first equation to form a new equation:\[x^2 + (x + 7)^2 = 13^2\]

Step 4: Simplify and solve the equation for x.

Expanding and simplifying the equation, we get:\[x^2 + (x^2 + 14x + 49) = 169\] Combining the terms, we have:\[2x^2 + 14x - 120 = 0\] Divide the equation by 2 to simplify:\[x^2 + 7x - 60 = 0\] Factor the quadratic equation:\[(x + 12)(x - 5) = 0\] Now find the values of x:\[x = -12, 5\]

Step 5: Determine the valid solution.

Since the distance from the bottom of the ladder to the wall cannot be negative, the valid solution is:\[x = 5\] So, the distance from the bottom of the ladder to the wall is 5 ft.

Key Concepts

Right-angled triangle

A right-angled triangle is a triangle in which one of the angles measures exactly 90 degrees. This type of triangle has a special property – the longest side, known as the hypotenuse, is always opposite the right angle. In our scenario, the ladder acts as the hypotenuse.

The two sides that form the right angle are called the legs of the triangle. In the ladder problem, these are:

• The distance from the bottom of the ladder to the wall (one leg).
• The distance from the top of the ladder to the bottom of the wall (second leg).

Understanding the configuration of this triangle is crucial for setting up equations using the Pythagorean theorem. This theorem connects the three sides of the right-angled triangle in a simple formula that allows for easy calculation of one side when the other two are known.

Quadratic equation

A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\). In this context, it emerges when we use variables to represent unknown quantities in our right-angled triangle problem. These equations often have two solutions, which we determine by techniques such as factoring, completing the square, or using the quadratic formula.

In the ladder problem, by substituting and expanding the expressions, we derived a quadratic equation:

• \(x^2 + 7x - 60 = 0\)

This equation was solved by factoring it into \((x + 12)(x - 5) = 0\). Here, the solutions to the equation, \(x = 5\) and \(x = -12\), represent potential real values for the distance. However, only the positive solution makes sense in the context of the physical problem.

Algebraic problem solving

Algebraic problem solving is about utilizing algebraic methods to solve equations that arise in real-world problems. This involves recognizing patterns and forming equations that make it easier to manage complex problems systematically.

For example, in the ladder scenario, we first used the Pythagorean theorem to connect the sides of the triangle. Then, we incorporated additional information given in the problem to form a system of equations. By substituting the expressions and simplifying, we solved the quadratic equation to find the distance.

• Identify what you know and what you need to find.
• Translate the problem into mathematical equations.
• Use algebraic manipulation to solve for unknowns.

These steps are essential in algebraic problem solving and ensure a structured approach to finding the solution.