Problem 70 - Introductory Algebra for College Students

Question

Use the Pythagorean Theorem to solve Exercises \(67-72\). Express the answer in radical form and simplify, if possible. The base of a 20 -foot ladder is 15 feet from the house. How far up the house does the ladder reach? (THE IMAGES CANNOT COPY)

Step-by-Step Solution

Answer

The ladder reaches 5 times the square root of 7 (~ 13.23) feet up the house.

Step 1: Identify the values in the problem

Identify the given lengths and assign them to their respective variables based on the Pythagorean theorem – in this case, c equals 20 feet, and b equals 15 feet. The exercise asks for the value of a.

Step 2: Apply the Pythagorean theorem

Use the rearranged Pythagorean theorem, \(a = \sqrt{c^2 - b^2}\), to calculate the height that the ladder reaches the house. Inserting the provided lengths (c and b), the equation is \(a = \sqrt{20^2 - 15^2}\).

Step 3: Calculate the value of a

Solve the equation to get the value of a. Doing the math, \(a = \sqrt{400 - 225} = \sqrt{175}\). The square root of 175 can be simplified by factoring out the perfect square 25 from 175 to obtain \(a = \sqrt{7*25} = 5\sqrt{7}\).

Key Concepts

Understanding Radical Form

The concept of expressing numbers in radical form is quite useful, especially when dealing with roots in geometry problems. Radical form allows us to express a number as a product that includes the square root symbol, which helps in preserving exact values rather than approximating with decimals.
If you're dealing with a number like 175 in radical form, it becomes crucial to search for perfect squares that can simplify the expression. To express 175 in radical form, we first recognize that it's composed of factors, particularly 5 and 7, multiplied by 25, which is a perfect square. By extracting the perfect square, we obtain the simplified radical form: \( \sqrt{175} = 5\sqrt{7} \).
This form keeps the expression exact and makes calculations more straightforward when solving geometry problems like using the Pythagorean Theorem.

Simplifying Square Roots

When simplifying square roots, the main goal is to find and extract perfect square factors. This means looking for numbers like 4, 9, 16, 25, etc., which are themselves squares of integers, within the radicand (the number under the square root symbol).
Let's take our example, \( \sqrt{175} \). We can break down 175 into its factors: 175 = 25 \(\times\) 7. Here, 25 is a perfect square, as it is 5 squared. We can then write \( \sqrt{175} \) as \( \sqrt{25 \times 7} \), which simplifies to \( 5\sqrt{7} \).

• Identify the largest perfect square factor of the radicand.
• Use the property \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \) to extract the square root of the perfect square.
• Simplify further if possible to make calculations easier.

This method helps keep our calculations neat and precise, avoiding unnecessarily complex or rounded numbers.

Basics of Triangle Geometry

Triangle geometry is a fundamental part of understanding and solving problems involving shapes and distances. The Pythagorean Theorem is particularly important for right triangles, where it relates the lengths of the three sides.
For any right triangle, the theorem states: \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, the side opposite the right angle, and \( a \) and \( b \) are the other two sides. In our ladder problem, the ladder itself is the hypotenuse, at 20 feet, while the distance from the house to the ladder's base is one leg of the triangle, at 15 feet. The other leg is the height up the house we're trying to find.
By rearranging the theorem for our scenario, we find \( a \) using \( a = \sqrt{c^2 - b^2} \). This rearrangement emphasizes how the theorem provides a reliable method to solve for any unknown side, making it invaluable in both theoretical and practical applications, like determining distances and heights in architecture or navigation, where exact measurements are vital.