Question

A wall 27 ft. high is 64 ft. from a house. Find the length of the shortest ladder that will reach the house if one end rests on the ground outside the wall.

Short answer

Answer: The length of the shortest ladder that will reach the house is approximately 69.47 feet.

Step-by-step solution

Identify the triangle sides

In this problem, we have a right triangle with one side measuring 27 ft (height of the wall) and the other side measuring 64 ft (distance between the wall and the house). We are tasked with finding the length of the ladder, which is the hypotenuse of the triangle.

Use the Pythagorean theorem

According to the Pythagorean theorem, the sum of the squares of the lengths of the two shorter sides of a right triangle is equal to the square of the length of the hypotenuse (longest side). The formula is given as follows: \(a^2 + b^2 = c^2\), where a and b are the two shorter sides of the triangle, and c is the hypotenuse. In this problem, we have a = 27 ft and b = 64 ft. We will substitute these values into the Pythagorean theorem and solve for c.

Substitute the given side lengths

Replace a with 27 and b with 64 in the formula \(a^2 + b^2 = c^2\): \((27)^2 + (64)^2 = c^2\)

Calculate the squares of the side lengths

Calculate the squares of the two side lengths: \(729 + 4096 = c^2\)

Add the squares of the side lengths

Add the squares of the side lengths: \(729 + 4096 = 4825\)

Solve for c (the hypotenuse)

We have \(4825 = c^2\). To solve for c, take the square root of both sides of the equation: \(c = \sqrt{4825}\)

Calculate the length of the ladder

Calculate the length of the ladder (c): \(c = \sqrt{4825} \approx 69.47\) ft The length of the shortest ladder that will reach the house if one end rests on the ground outside the wall is approximately 69.47 feet.

Key concepts

Right Triangles

When dealing with right triangles, it's essential to understand their structure. A right triangle is a triangle in which one angle measures exactly 90 degrees. This characteristic makes the right triangle unique and sets the stage for special rules that apply to it.

Knowing the three parts of a right triangle is crucial:

• The two sides that form the right angle are called the 'legs.'
• The side opposite the right angle is known as the 'hypotenuse.'
• The hypotenuse is always the longest side in a right triangle.

Right triangles are not just limited to academic problems. They have practical applications in various areas, such as construction, navigation, and even art. Understanding their properties can greatly aid in solving both theoretical and real-world problems.

Hypotenuse Calculation

Calculating the hypotenuse is a common task in problems involving right triangles. To find the hypotenuse, the Pythagorean theorem is your best ally. This fantastic theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). The formula can be neatly written as:

• \( a^2 + b^2 = c^2 \)

By inserting the known values of the legs into this formula, you can solve for the unknown hypotenuse. For example, in the problem we have, with a wall 27 ft high (a) and a distance of 64 ft to the house (b), substituting into the formula gives:

\((27)^2 + (64)^2 = c^2\)

Calculating gives \(729 + 4096 = c^2\), and hence \(c^2 = 4825\). To find the hypotenuse, take the square root of 4825, resulting in approximately 69.47 feet.

Mathematical Problem Solving

Mathematical problem solving requires a blend of analytic and strategic thinking. When faced with a problem like finding the shortest ladder to reach a house over a wall, breaking the problem into manageable steps makes it easier.

Begin by identifying all elements of the problem:

• Recognize it as a right triangle problem.
• Identify the given measurements and what needs to be found.

The next step is to apply the appropriate theorem or rule, such as the Pythagorean theorem in this case, allowing you to solve for the unknown. By substituting the known values into the theorem, you plug away at the calculations until you arrive at the solution.

Throughout the process, it's crucial to check your work. Make sure calculations are accurate, as small mistakes can lead to incorrect conclusions. Using estimation for verification helps ensure your solution is reasonable. Engaging in such systematic steps enhances not just your mathematical ability but also critical thinking skills, which are beneficial across various problem-solving scenarios.