Problem 11 - Technical Mathematics with Calculus

Question

To help you solve each problem, draw a diagram and label it completely. Look for special triangles or right triangles contained in the diagram. Be sure to look up any word that is unfamiliar. A ladder 39.0 ft long reaches to the top of a building when its foot stands \(15.0 \mathrm{ft}\) from the building. How high is the building?

Step-by-Step Solution

Answer

The height of the building is 36.0 ft.

Step 1: Draw the Diagram

Sketch a right triangle to represent the scenario. The ladder forms the hypotenuse, the distance from the ladder's foot to the building is one leg, and the height of the building is the other leg.

Step 2: Label the Diagram

Label the hypotenuse as 39.0 ft (ladder's length), the base as 15.0 ft (distance from the ladder's foot to the building), and the height as 'h' (the unknown we need to find).

Step 3: Apply the Pythagorean Theorem

Use the Pythagorean Theorem (\(a^2 + b^2 = c^2\)) to solve for the height 'h', where 'c' is the hypotenuse (39.0 ft), and 'a' is the base (15.0 ft).

Step 4: Solve for the Height

Plug the values into the Pythagorean Theorem equation, getting \(15.0^2 + h^2 = 39.0^2\) and solve for 'h'.

Step 5: Calculate the Height

Calculate the unknown leg 'h' by rearranging the formula: \(h^2 = 39.0^2 - 15.0^2\). Then, take the square root of both sides to solve for 'h'.

Step 6: Evaluate the Height

Compute the values to find \(h^2 = 1521 - 225\) which simplifies to \(h^2 = 1296\). Taking the square root gives us \(h = 36.0 \mathrm{ft}\).

Key Concepts

Right Triangle

A right triangle is a powerful figure in geometry and an essential building block in trigonometry. It is characterized by one angle measuring exactly 90 degrees, which makes it unique when compared to other triangles. Because of this right angle, a right triangle has distinctive properties that allow for the application of the Pythagorean Theorem.

In our example, the ladder leaning against the building creates a right triangle with the ground and the building. The ladder itself represents the hypotenuse, the longest side opposite the right angle, while the distance from the ground to the building and the building's height represent the other two sides, also known as the 'legs' of the triangle.

Understanding the relationship between the sides of a right triangle is crucial for not only solving direct problems involving the triangle itself but also for comprehending more advanced mathematics and physics concepts where right triangles are used indirectly to solve problems.

Trigonometry

Trigonometry is the study of the relationships between the angles and sides of triangles, especially right triangles. It extends beyond the scope of the Pythagorean Theorem to include other aspects such as the sine, cosine, and tangent ratios which provide further insight into angle measures and side lengths.

For simpler cases, such as the ladder problem, trigonometry might seem like an overcomplication. However, understanding trigonometric functions is crucial when dealing with angles and side measurements that are not as straightforward as in the current scenario. Trigonometry will be able to tell us, for example, the angles at which the ladder is leaning against the building or how much shorter the ladder can be if it leans at a steeper angle while still reaching the same height on the building.

Mathematical Problem-Solving

Solving mathematical problems often requires a step-by-step approach that breaks down a problem into manageable parts. This process typically starts with understanding the problem, devising a plan, implementing the plan, and finally reviewing the solution.

Our exercise demonstrated this with the ladder problem: understanding was achieved through recognizing the presence of a right triangle; planning involved labeling the sides and applying the Pythagorean Theorem; implementation was carried out through algebraic manipulation; and reviewing ensured that our solution was reasonable given the context. With these steps, students can tackle a wide array of problems systematically, reducing the likelihood of errors and building confidence in their problem-solving abilities.

Technical Mathematics

Technical mathematics refers to the application of mathematical concepts to solve problems in technical fields like engineering, physics, computer science, and more. Here, precision and an understanding of real-world implications are paramount.

In the context of the ladder problem, technical mathematics would consider additional factors such as the stability of the ladder, the angle at which it's safe to place the ladder, and the strength of the materials involved—issues that extend the problem beyond theoretical calculation into practical application. For students aspiring to technical careers, mastery in these concepts opens doors to advanced study and real-world problem-solving.