Problem 102 - Algebra and Trigonometry

Question

Reach of a Ladder \(A 19 \frac{1}{2}\) -foot ladder leans against a building. The base of the ladder is 7\(\frac{1}{2}\) ft from the building. How high up the building does the ladder reach?

Step-by-Step Solution

Answer

The ladder reaches 18 feet up the building.

Step 1: Identify the Problem

We need to find out how high the ladder reaches on the building. This problem can be solved using the Pythagorean theorem, where the ladder forms the hypotenuse of a right triangle, the distance from the building is one leg, and the height reached on the building is the other leg.

Step 2: Assign Values and Understanding of Terms

Let the length of the ladder be represented by the hypotenuse, which is 19.5 feet, and the distance from the building, one leg of the right triangle, be 7.5 feet. We need to find the height, represented as the other leg of the triangle, using the Pythagorean theorem formula.

Step 3: Write the Pythagorean Theorem

The Pythagorean Theorem formula is: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the legs of the triangle, and \( c \) is the hypotenuse.

Step 4: Plug in Known Values

Using the formula, set \( a = 7.5 \) ft (base) and \( c = 19.5 \) ft (hypotenuse). We need to find \( b \) (the height): \[ (7.5)^2 + b^2 = (19.5)^2 \]

Step 5: Calculate Squares of Known Values

Calculate the squares: \((7.5)^2 = 56.25\) and \((19.5)^2 = 380.25\). Update the equation to: \[ 56.25 + b^2 = 380.25 \]

Step 6: Solve for the Unknown

Subtract 56.25 from both sides to solve for \( b^2 \): \[ b^2 = 380.25 - 56.25 \] \[ b^2 = 324 \]

Step 7: Find the Square Root

Find \( b \) by taking the square root of 324: \[ b = \sqrt{324} \] \[ b = 18 \]

Step 8: Conclusion

The ladder reaches a height of 18 feet up the building.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that includes one angle measuring exactly 90 degrees. This specific angle creates a variety of interesting mathematical properties. Within these triangles, the side opposite the right angle is called the hypotenuse, while the other two sides, which meet at the right angle, are referred to as legs.
The solution to many right triangle problems, including those using ladders leaning against buildings, involves recognizing the relationship between these sides. In applied problems, modeling these scenarios as right triangles allows the use of the Pythagorean theorem to find missing side lengths effectively and accurately.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is always opposite the right angle. In practical terms, if you imagine a ladder leaning against a wall, the ladder itself acts as the hypotenuse of the right triangle formed between the ladder, the ground, and the wall.
When solving problems involving the hypotenuse, it is crucial to ensure the ladder's length is accurately measured since it plays a key role in calculations. Knowing the length of the hypotenuse, along with one leg of the triangle, allows for determining the other leg using the Pythagorean theorem.

Squares

In the context of right triangles and the Pythagorean Theorem, squares refer to the result of multiplying a number by itself. This operation is denoted by the exponent 2. For instance, calculating the square of a number like 7.5 involves computing 7.5 x 7.5, which equals 56.25.
In the Pythagorean theorem, each leg's length is squared and summed. For our ladder problem, squaring involves taking known lengths (such as the base or the hypotenuse) and transforming them into squared values, helping to equate them more straightforwardly within the theorem's equation.

Square Root

A square root is the inverse operation of squaring a number. It involves finding a number which, when multiplied by itself, yields the original square. For example, the square root of 324 is 18 because 18 x 18 equals 324.
In solving the ladder problem, after finding the equation for the unknown height squared, the next step was taking the square root of the result to find the actual height. By finding the square root of 324, we determine the ladder can safely reach 18 feet up the building.