Problem 141 - Algebra and Trigonometry

Question

Use the Pythagorean Theorem and the square root property to solve. Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. The base of a 30 -foot ladder is 10 feet from a building. If the ladder reaches the flat roof, how tall is the building?

Step-by-Step Solution

Answer

The height of the building is \(20\sqrt{2}\) feet, or approximately 28.3 feet when rounded to the nearest tenth.

Step 1: Understand and Setup the Pythagorean Theorem

The Pythagorean Theorem can be represented as \(a^2 + b^2 = c^2\), where c is the hypotenuse, and a and b are the two sides of the triangle. In this example, the hypotenuse (c) is the ladder which measures 30 feet, 'a' is the distance of the ladder from the wall which is 10 feet and 'b' is the height of the building which we need to calculate.

Step 2: Substitute into the Pythagorean Theorem

Substitute the given values into the Pythagorean theorem equation. So, you would have \(10^2 + b^2 = 30^2\).

Step 3: Simplify the Equation

Simplify this equation. \(b^2 = 30^2 - 10^2 = 900 - 100 = 800\).

Step 4: Solve for the Building Height

To find 'b', take the square root of both sides of the equation. \(b = \sqrt{800}\)

Step 5: Simplify the Square Root

The square root of 800 simplifies to \(20\sqrt{2}\), which is the exact value for the building height. Then to find the decimal approximation to the nearest tenth, calculate \(20\sqrt{2}\) which is approximately 28.3 feet.

Key Concepts

Square Root Property

The square root property is a valuable mathematical tool, especially when dealing with the sides of a right-angled triangle. It involves taking the square root of both sides of an equation to solve for a variable. In our problem, if you have an equation like \(b^2 = 800\), the square root property allows you to solve for \(b\) by applying \(b = \sqrt{800}\). This step is crucial as it transforms the equation from a squared term to its root, making it easier to solve for the unknown variable.
This method is particularly useful because squares often appear in geometric problems, especially those involving the Pythagorean Theorem. Doing this gives us the length of one side of the right triangle when other side lengths are known.
Use this method whenever you need to isolate a term that has been squared.

Simplified Radical Form

Simplified radical form is the process of expressing a square root in its simplest form. It ensures that the answer is as clear and straightforward as possible. In our example, after finding \(b = \sqrt{800}\), we further simplify this expression. By finding the largest square factor of 800, which is 400, we can break it down as \(\sqrt{800} = \sqrt{400 \times 2} = \sqrt{400} \times \sqrt{2}\).
Since the square root of 400 is 20, we simplify \(\sqrt{800}\) to \(20\sqrt{2}\). This is the simplified radical form of \(b\).
Such simplification is important as it provides the most concise and accurate depiction of a radical expression, free of any perfect square factors within the radical sign, which can also be very useful in further calculations.

Decimal Approximation

Decimal approximation is a way of finding a more accessible but less exact representation of numbers, particularly when dealing with irrational numbers like \(20\sqrt{2}\). After you've simplified to the radical form, you may need to give an answer that is more practical for everyday use or when you need precision to a specific decimal place.
This is achieved by calculating \(20\sqrt{2}\), using a calculator or other computational tool. This calculation approximates \(20\sqrt{2}\) to about 28.3 when rounded to the nearest tenth, which means that the height of the building is approximately 28.3 feet.
Decimal approximations are regularly used because they give a straightforward number that can be easily understood, providing a quick and practical reference to the magnitude of a number without the complexity of radical expressions.