Problem 142 - College Algebra Essentials

Question

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) 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 building is \(20\sqrt{2}\) feet tall which is approximately 28.3 feet to the nearest tenth

Step 1: Identify the known and unknown quantities

From the problem, it is known that the length of the ladder(which is the hypotenuse of the triangle) is 30 feet and the distance of the ladder base from the building(base of the triangle) is 10 feet. The height of the building is the unknown quantity that needs to be found.

Step 2: Apply the Pythagorean theorem

As per the Pythagorean theorem, in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Therefore, \(height^2 = ladder^2 - base^2 = 30^2 - 10^2 = 900 - 100 = 800\). Hence, the height of the building in square feet equals 800.

Step 3: Find the exact height using square root

To find the exact height of the building, take the square root of the height squared. Therefore, \(height = \sqrt{800} = 20\sqrt{2}\) feet. This is the simplified radical form of the height.

Step 4: Find the decimal approximation to the nearest tenth

To find a decimal approximation of the height to the nearest tenth: The square root of 800 is approximately 28.3. Therefore, the height of the building is approximately 28.3 feet when rounded to the nearest tenth.

Key Concepts

Understanding Right Triangles

Right triangles are special types of triangles that have one angle measuring exactly 90 degrees. This right angle is crucial because it allows for specific mathematical properties and theorems to apply, such as the Pythagorean Theorem.

• Each right triangle consists of two shorter sides called "legs" and one longer side called the "hypotenuse."
• The hypotenuse is directly opposite the right angle, and it's always the longest side in a right triangle.
• Problems involving right triangles often require finding one of these sides when given the other two.

Understanding these components is essential because they form the basis of many geometric and trigonometric calculations in mathematics.
This understanding helps when using the Pythagorean Theorem to find unknown side lengths.

Applying the Square Root Property

The square root property is a tool used to solve equations involving squared terms. This property can help find the length of a side in a right triangle when using the Pythagorean Theorem. In the given exercise:

• The height squared (u) was solved to be 800 from the equation setup: \(height^2 = ladder^2 - base^2 \).
• To isolate the height, the next step is to take the square root of both sides: \(height = \sqrt{800} \).

This step is crucial as it transforms the problem from dealing with squared numbers back to measuring actual lengths.
It’s important to know that taking square roots can produce either positive or negative results, but in geometry, especially in measuring real lengths such as the height of a building, we only consider the positive square root.
By understanding this property, solving equations with square terms becomes more routine.

Expressing Answers in Simplified Radical Form

Simplified radical form is often preferred in mathematics for its accuracy and beauty. A radical, such as \(\sqrt{800}\), can often be expressed in a simpler form. To convert a radical into its simplest form, you need to factor the number inside the radical:

• Notice that 800 can be broken down into smaller factors: \(800 = 16 \times 50\).
• Since \(16\) is a perfect square, its square root can be simplified: \(\sqrt{16} = 4\).
• This simplifies \(\sqrt{800}\) to \(4 \times \sqrt{50}\).
• Continue factoring: \(50 = 25 \times 2\), where \(\sqrt{25} = 5\).
• The final simplified form is thus \(20 \sqrt{2}\).

Expressing length in simplified radical form like \(20 \sqrt{2}\) provides precise, compact expression. It’s precise enough for most theoretical math work. Simplified form will often reveal further calculations more straightforwardly than decimal or fractional forms.

Finding Decimal Approximations

When working with radicals or other mathematical expressions, sometimes a numerical estimate is more useful, especially in practical scenarios. This is where decimal approximation comes into play.
To find the decimal approximation of our previous expression \(20 \sqrt{2}\):

• Use a calculator to determine the value of \(\sqrt{2}\), which is approximately 1.414.
• Multiply this value by 20 to get 28.28.

When rounded to the nearest tenth, this gives us a height of approximately 28.3 feet.
Decimal approximations are incredibly useful for quick estimates and real-world applications where precision to the exact decimal is less critical.
Remember, while it provides a quick glance, a decimal approximation loses some mathematical precision compared to keeping the answer in radical form.