Problem 49 - Precalculus Mathematics for Calculus

Question

A \(19 \frac{1}{2}\) -foot ladder leans against a building. The base of the ladder is \(7 \frac{1}{2} \mathrm{ft}\) 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: Understand the Problem

The problem involves a right-angled triangle formed by the ladder, the wall, and the ground. The ladder is hypoteneuse, distance from wall to base is one leg, and height is another leg.

Step 2: Identify Known Values

We know the length of the ladder (hypotenuse) is \( 19 \frac{1}{2} \) feet, which is equal to \( \frac{39}{2} \) feet, and the distance from the wall to the base of the ladder (one leg) is \( 7 \frac{1}{2} \) feet, which is equal to \( \frac{15}{2} \) feet.

Step 3: Apply the Pythagorean Theorem

Use the Pythagorean theorem, which is \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides of the triangle.

Step 4: Substitute the Known Values

Substitute the given values into the equation: \( \left( \frac{15}{2} \right)^2 + b^2 = \left( \frac{39}{2} \right)^2 \).

Step 5: Simplify the Equation

Calculate \( \left( \frac{15}{2} \right)^2 = \frac{225}{4} \) and \( \left( \frac{39}{2} \right)^2 = \frac{1521}{4} \). Substitute back into the equation to get \( \frac{225}{4} + b^2 = \frac{1521}{4} \).

Step 6: Solve for \( b^2 \)

Rearrange the equation to solve for \( b^2 \): \( b^2 = \frac{1521}{4} - \frac{225}{4} = \frac{1296}{4} \).

Step 7: Calculate \( b \)

Find \( b \) by taking the square root of \( \frac{1296}{4} \): \( b = \sqrt{324} = 18 \).

Step 8: Conclude the Solution

Therefore, the ladder reaches a height of 18 feet up the building.

Key Concepts

Right-Angled Triangle

A right-angled triangle is a type of triangle that includes one angle measuring exactly 90 degrees. This angle is called the right angle and it serves as the foundation for this category of triangles.

In our problem, the ladder, building, and ground form a right-angled triangle. The ground and building form the two legs of the triangle.

Since these two legs meet at a 90-degree angle, they provide the right-angled corner we need to define our triangle.

Hypotenuse

The hypotenuse is a special side of a right-angled triangle. It is always the longest side because it lies opposite the right angle.

In this scenario, the ladder acts as the hypotenuse. With a given length of \(19 \frac{1}{2}\) feet, it is known to be longer than the other sides of the triangle.

One useful property of the hypotenuse and right-angled triangles is that they allow us to use the Pythagorean Theorem to find missing lengths, a key aspect of this problem.

Solve for Unknown Side

To find the unknown side in a right-angled triangle, the Pythagorean Theorem is the perfect tool. The theorem is represented by the equation \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.

In this exercise, we need to find the height the ladder reaches up the building, which is one of the legs. We know:

• The hypotenuse (ladder) is \( \frac{39}{2} \) feet.
• The base from the wall to the ladder is \( \frac{15}{2} \) feet.

Substitute these values into the Pythagorean Theorem. Solve for the unknown side by rearranging and simplifying the equation step by step.

Ultimately, take the square root of your final equation for \(b^2\) to find \(b\). This will give you the exact height reached by the ladder.

Mathematical Problem Solving

Mathematical problem solving is a skill that involves logical thought and step-by-step processing. This exercise demonstrates several essential problem-solving steps:

• Understand the problem by visualizing or sketching the situation if necessary.
• Identify known and unknown values to organize what information you have and what you need to find out.
• Choose a strategy, such as an appropriate mathematical formula or theorem, like the Pythagorean Theorem.
• Execute the solution by carefully substituting values, performing arithmetic, and simplifying expressions while checking your work.
• Verify the result by ensuring it fits the context of the problem.

Through this process, a mathematical solution emerges, helping us find the height the ladder reaches up the building.