Problem 25 - Prealgebra

Question

Height of a Tree A tree casts a shadow 38 feet long, while a 6 -foot man casts a shadow 4 feet long. How tall is the tree?

Step-by-Step Solution

Answer

The tree is 57 feet tall.

Step 1: Understand the Problem

We know that the length of the shadow cast by any object is proportional to its height when the sun is shining at the same angle. We need to find the height of the tree given the length of its shadow and the height and shadow length of the man.

Step 2: Set Up the Proportion

Since the tree and the man are casting shadows under the same conditions, we can set up a proportion using their respective heights and shadow lengths. Let the height of the tree be denoted by \( h \). The proportion is:\[ \frac{h}{38} = \frac{6}{4} \]

Step 3: Solve the Proportion

Cross-multiply to solve the proportion for the tree's height \( h \):\[ 4h = 6 imes 38 \]Calculating gives:\( 4h = 228 \).

Step 4: Calculate the Height of the Tree

Divide both sides of the equation by 4 to find \( h \):\[ h = \frac{228}{4} \]This gives:\( h = 57 \).

Step 5: Verify the Solution

Recheck the calculations to ensure accuracy. The provided proportion maintains that if a 6-foot man casts a shadow of 4 feet, then a 57-foot tree should cast a shadow of 38 feet. The calculations align correctly.

Key Concepts

Understanding Shadow Problems

Shadow problems are a classic type of mathematical exercise often involving proportions. They help us understand how shadows, created by the same light source, can provide valuable information about the heights or lengths of different objects. When two objects, like a tree and a man, are subjected to the same light conditions, their shadows allow us to form a ratio between their heights and shadow lengths.

These problems serve as practical examples of using proportions in real-world scenarios. They are particularly useful for realizing how similar triangles form when light sources, like the sun, cast shadows on various objects. This understanding lays the groundwork for setting up and solving proportions effectively.

Cross-Multiplication for Proportions

Cross-multiplication is a powerful technique to solve proportions. In a proportion, we equate two ratios, and by cross-multiplying, we can find an unknown term in the equation easily. This method is crucial because it simplifies the process of dealing with fractions and helps us isolate the variable we need to find.

For instance, in this exercise, the proportion \[ \frac{h}{38} = \frac{6}{4} \] is used. By cross-multiplying, we multiply across the equals sign diagonally: \[ 4h = 6 \times 38 \] which results in \[ 4h = 228 \]. Cross-multiplication helps us move from a proportion to a simple algebraic equation, making it more straightforward to solve for the unknown variable.

Effective Problem-Solving Steps

Good problem-solving relies on a structured approach. Here, breaking down the shadow problem into clear steps aids in understanding and solving it efficiently. The first step is to comprehend the problem—identify what is given and what needs to be found. Recognizing that shadow lengths are proportional to the objects casting them allows us to set up a useful proportion.

Once the proportion is set up, we can use cross-multiplication to solve for the unknown, following a linear process of calculations. Double-checking is critical as the final step to ensure that our calculations align with the problem's conditions. This series of logical steps not only simplifies the task but builds confidence in problem-solving ability.

Applying Mathematical Reasoning

Mathematical reasoning involves applying logical thinking to solve problems. In shadow problems, this means using the understanding of proportionate relationships and effectively manipulating equations to find solutions. Reasoning helps us predict the kind of relationship objects will share under the same lighting conditions.

By establishing that a 6-foot man and a towering tree both abide by the same shadow conditions, we draw logical conclusions on the unknown height of the tree. Logical deductions combined with algebraic manipulation help us arrive at the solution: a 57-foot tree.

Thus, mathematical reasoning bridges the gap between abstract numbers and real-world interpretations, turning a simple observation into a valued solution.