Problem 36 - Beginning and Intermediate Algebra

Question

Solve each problem. When appropriate, round answers to the nearest tenth. A 13-ft ladder is leaning against a house. The distance from the bottom of the ladder to the house is \(7 \mathrm{ft}\) less than the distance from the top of the ladder to the ground. How far is the bottom of the ladder from the house?

Step-by-Step Solution

Answer

The bottom of the ladder is 5 ft from the house.

Step 1 - Define Variables

Let the distance from the bottom of the ladder to the house be denoted as x ft. Since the distance from the bottom of the ladder to the house is 7 ft less than the distance from the top of the ladder to the ground, we can represent the length from the top of the ladder to the ground as x + 7 ft.

Step 2 - Apply the Pythagorean Theorem

Since the ladder, house, and ground form a right triangle, we can use the Pythagorean Theorem: \( a^2 + b^2 = c^2 \), where c is the hypotenuse (ladder), a is the height, and b is the base. Here, the ladder is 13 ft, and the sides are x and x + 7 ft. Hence, we write:\[ x^2 + (x + 7)^2 = 13^2 \]

Step 3 - Simplify the Equation

Expand the equation to simplify:\[ x^2 + (x + 7)^2 = 169 \]\[ x^2 + x^2 + 14x + 49 = 169 \]Combine like terms:\[ 2x^2 + 14x + 49 = 169 \]

Step 4 - Form a Quadratic Equation

To create a standard quadratic equation, subtract 169 from both sides:\[ 2x^2 + 14x + 49 - 169 = 0 \]\[ 2x^2 + 14x - 120 = 0 \]

Step 5 - Simplify the Quadratic Equation

Divide the entire equation by 2 to make it simpler:\[ x^2 + 7x - 60 = 0 \]

Step 6 - Solve the Quadratic Equation

Solve the quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where a = 1, b = 7, and c = -60:\[ x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 1 \cdot (-60)}}{2 \cdot 1} \]\[ x = \frac{-7 \pm \sqrt{49 + 240}}{2} \]\[ x = \frac{-7 \pm \sqrt{289}}{2} \]\[ x = \frac{-7 \pm 17}{2} \]This gives two solutions: \( x = 5 \) and \( x = -12 \). Since distance cannot be negative, we take the positive value:\[ x = 5 \]

Step 7 - Verify the Solution

Substitute x = 5 back into the original relationship to verify:The distance from the top of the ladder to the ground is x + 7 = 12 ft. Using the Pythagorean theorem \[ 5^2 + 12^2 = 13^2 \]\[ 25 + 144 = 169 \]\[ 169 = 169 \]The solution is verified.

Key Concepts

Algebra

In the given problem, algebra plays a crucial role in setting up the relationship between different lengths of the ladder and solving for unknown variables. We start by defining a variable for the unknown distance from the bottom of the ladder to the house.

Here's a quick recap: Let the distance from the bottom of the ladder to the house be denoted as x ft. Since this distance is 7 ft less than from the top of the ladder to the ground, we can write the height as x + 7 ft.

Using algebra, we set up an equation involving these variables. This step is essential as it translates the word problem into mathematical terms that can be solved systematically. Recognizing and defining variables in algebra helps break down complex real-world problems into simpler, manageable parts.

Right Triangle

The problem involves a right triangle formed by the ladder, the wall of the house, and the ground. Understanding the properties of right triangles is fundamental to solving this problem.

A right triangle has one 90-degree angle. The sides are called the legs, and the longest side opposite the right angle is the hypotenuse. In this problem:

• The ladder acts as the hypotenuse (13 ft).
• One leg is the distance from the bottom of the ladder to the house (x ft).
• The other leg is from the top of the ladder to the ground (x + 7 ft).

We use the properties of right triangles to apply the Pythagorean Theorem, which relates the lengths of the sides in these triangles.

Quadratic Equation

To solve the problem, we converted the Pythagorean Theorem equation into a quadratic equation. Here’s the detailed process:

We started with:
\[x^2 + (x + 7)^2 = 169\]

Expanding and simplifying this, we obtained:

\[2x^2 + 14x + 49 = 169\]
Then, we formed the standard quadratic equation:

\[x^2 + 7x - 60 = 0\]

Solving this required applying the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Substituting a = 1, b = 7, and c = -60 gave us the solutions. The process of solving quadratic equations helps us find the possible values for x, enabling us to determine the distance we seek.

Distance Calculation

Finally, we verified the solution by checking the calculated distances using the Pythagorean Theorem.

We had two possible solutions (x = 5 and x = -12), but only the positive value makes sense. Substituting x = 5 back into our initial equation:

• The distance from bottom of the ladder to the house is 5 ft.
• The distance from the top of the ladder to the ground is 5+7=12 ft.

Using the Pythagorean Theorem to verify:
\[5^2 + 12^2 = 13^2\]
\[25 + 144 = 169\]
\[169 = 169\]
The distances satisfy the relationship confirming that the bottom of the ladder is indeed 5 feet away from the house.