Problem 41 - Technical Mathematics with Calculus

Question

Project: In The Musgrave Ritual, Sherlock Holmes calculates the length of the shadow of an elm tree that is no longer standing. He does know that the elm was 64 ft high and that the shadow was cast at the instant that the sun was grazing the top of a certain oak tree. Holmes held a \(6-\) ft-long fishing rod vertical and measured the length of its shadow at the proper instant. It was 9 ft long. He then said, "Of course the calculation now was a simple one. If a rod of six feet threw a shadow of nine, a tree of sixty-four feet would throw one of _____ "How long was the shadow of the elm?

Step-by-Step Solution

Answer

The shadow of the elm tree was 96 feet long.

Step 1: Identify the Similar Triangles

The situation describes two similar triangles, with the fishing rod and its shadow forming one triangle, and the elm tree and its shadow forming a corresponding larger triangle. In both cases, the height of the object is perpendicular to the ground, and the length of the shadow forms the base of the triangle.

Step 2: Write the Proportion

By the principles of similar triangles, the ratio of the height of the fishing rod to the length of its shadow will equal the ratio of the height of the elm tree to the length of its shadow. This is expressed as: \(\frac{height_{rod}}{shadow_{rod}} = \frac{height_{elm}}{shadow_{elm}}\).

Step 3: Plug in Known Values

Substitute the known values into the proportion: \(\frac{6 \text{ ft}}{9 \text{ ft}} = \frac{64 \text{ ft}}{shadow_{elm}}\).

Step 4: Solve for the Unknown

To find the length of the shadow of the elm, solve the equation for \(shadow_{elm}\): \(shadow_{elm} = \frac{64 \text{ ft} \times 9 \text{ ft}}{6 \text{ ft}}\).

Step 5: Calculate the Result

Perform the multiplication and division to find the shadow length of the elm: \(shadow_{elm} = \frac{576}{6} \text{ ft} = 96 \text{ ft}\). Thus, the shadow of the elm tree was 96 feet long.

Key Concepts

Proportion Calculation

Understanding proportion calculation is essential for solving various problems in mathematics, particularly in geometry. A proportion is an equation that states that two ratios are equal. It's an expression of the relationship between parts of a whole.

For example, if a fishing rod 6 feet tall casts a shadow of 9 feet when the sun is at a certain angle, and you want to find out how long the shadow would be for any other height, you use proportions. In the exercise concerning Sherlock Holmes, the proportion is set up by comparing the height of the rod and its shadow, to the height of the elm and its shadow. Once the proportion is created, we cross-multiply to solve for the unknown value, showing that this method is efficient and universally applicable to similar scenarios.

Similar Triangles in Trigonometry

Similar triangles play a pivotal role in trigonometry, helping to solve problems without requiring direct measurement. When triangles are similar, their corresponding angles are equal, and their sides are proportional. This characteristic enables us to use one triangle to make inferences about another.

Trigonometry often deals with the ratios of sides within right triangles to find unknown lengths or angles. Those ratios—sine, cosine, and tangent—are the foundation of many real-world applications like navigation, engineering, and physics. The exercise with Holmes' deduction about the elm's shadow incorporates the idea of similar triangles. The known dimensions from the rod’s triangle help determine the unknown length of the elm's shadow by applying these fundamental trigonometric principles.

Geometry Problem-Solving

Geometry problem-solving is a systematic way to find solutions using shapes, sizes, positions, and properties of space. In geometry, visualizing the problem is often key to finding the solution. By breaking down the exercise into manageable steps — identifying similar triangles, expressing the relationships through a proportion, and then solving for the unknown — we transform a seemingly complex problem into a straightforward calculation.

Effective problem-solving in geometry also entails familiarity with theorems and properties of shapes such as triangles, quadrilaterals, and circles. By taking Sherlock Holmes' approach of step-by-step reasoning, students not only reach the solution but also gain a deeper insight into how geometric theorems and properties are practically applied.

Mathematical Ratios

Ratios are a fundamental concept in mathematics, representing the quantitative relationship between two numbers. They indicate how many times one number contains another and are essential in comparing sizes, speeds, densities, and many more attributes.

Ratios are especially crucial when we do not have all the necessary information but have enough to infer unknown quantities through their relationships with known ones. In the context of our Sherlock Holmes example, the ratio of the fishing rod’s height to its shadow allows us to infer the length of the tree’s shadow by establishing a constant ratio. Mastery of mathematical ratios is a powerful tool in a student's skill set for solving various problems in algebra, geometry, and beyond.