Problem 71 - Elementary Algebra

Question

At the same time of day, a tree casts a 12 -foot shadow while a 6-foot man casts a 3-foot shadow. Estimate the height of th tree.

Step-by-Step Solution

Answer

The height of the tree is 24 feet.

Step 1: Identify the known values

We have two known sets of information: 1. A 6-foot man casts a 3-foot shadow. 2. A tree casts a 12-foot shadow. We need to use these sets of data to find the height of the tree.

Step 2: Apply the concept of similar triangles

The situation involves similar triangles because the sun's rays will strike both the man and the tree at the same angle. Thus, we can set up a proportion based on the heights and shadow lengths:\[ \frac{\text{Height of man}}{\text{Shadow of man}} = \frac{\text{Height of tree}}{\text{Shadow of tree}} \]

Step 3: Substitute the known values into the proportion

Substitute the known values into the proportion:\[ \frac{6}{3} = \frac{\text{Height of tree}}{12} \]

Step 4: Solve for the height of the tree

Cross-multiply to solve for the height of the tree:\[ 6 \times 12 = 3 \times \text{Height of tree} \]This simplifies to:\[ 72 = 3 \times \text{Height of tree} \]Divide both sides by 3:\[ \text{Height of tree} = \frac{72}{3} = 24 \]

Step 5: Conclusion

Based on the calculations, the height of the tree is 24 feet.

Key Concepts

Understanding Proportions

Proportions are a way of expressing a relationship between two ratios or fractions. They are especially useful when comparing similar figures, such as triangles. In our exercise, both the man and the tree create two pairs of similar triangles due to the constant angle of the sun's rays. A proportion is written as follows:\[ \frac{a}{b} = \frac{c}{d} \]Here, \( a \), \( b \), \( c \), and \( d \) are numbers or expressions. If these ratios are equal, we can say that they are in proportion to each other.

• Height of Man vs Shadow of Man: The ratio is \( \frac{6}{3} \).
• Height of Tree vs Shadow of Tree: The unknown ratio we need to find; \( \frac{\text{Height of Tree}}{12} \).

By establishing these proportions, we can solve for unknowns like the height of the tree. This relationship gives us the necessary framework to find missing values when analyzing geometric figures or measurements.

Calculating Shadow Measurements

Measuring shadows is a practical application that involves understanding the relationship between objects and their corresponding shadows. This concept is particularly useful in problems involving similar triangles, as shadows create one side of the triangle. To deduce measurements from shadows: - Observe the angles: When the sun is at a certain angle, it casts shadows differently. Both objects in our scenario will cast similar angles of shadow, forming similar triangles. - Check the context: The problem provides the lengths of the shadows for both the tree and the man. By knowing the length of one side (shadow) and the corresponding side on the similar triangle (height of the tree or man), we can use these measurements to solve the problem using proportions. In this case:

• The man's shadow is 3 feet, while the tree's shadow is 12 feet.
• This gives us a direct method to calculate proportions using the shadow's length.

This approach makes shadow measurement a straightforward method for estimating unknown attributes such as the height of an object.

Solving with Algebraic Equations

Algebra provides the tools needed to solve for unknown values using established equations. Once we have set up a proportion from similar triangles, algebra comes into play to find the value we need.In our exercise, the task was to find the height of the tree using the set proportion:\[ \frac{6}{3} = \frac{\text{Height of tree}}{12} \]Here's how we solve it:1. **Cross-Multiply:** This method involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. \[ 6 \times 12 = 3 \times \text{Height of tree} \] 2. **Simplify:** This gives \( 72 = 3 \times \text{Height of tree} \).3. **Solve for Tree Height:** Divide both sides by 3 to isolate the variable: \[ \text{Height of tree} = \frac{72}{3} = 24 \]Applying these algebraic principles helps in finding the exact height of the tree. Algebra effectively translates the relationship in proportions into straightforward calculation steps.