Question

On a sunny day, a tree casts a shadow that is 33 feet long. At the same time, a nearby 4-foot-tall mailbox casts a shadow that is \(5.5\) feet long. What is the height of the tree? Round your answer to the nearest foot. (A) 24 feet (B) 33 feet (C) 38 feet (D) 45 feet

Short answer

The height of the tree is \(24\) feet.

Step-by-step solution

Set up a proportion

We can set up a proportion using the heights and shadow lengths of both the tree and the mailbox. Let's use the variable \(h\) to represent the height of the tree: \[\frac{h}{33} = \frac{4}{5.5}\]

Solve for h

To solve for \(h\), cross-multiply: \(5.5h = 33 \cdot 4\) Now, we can divide both sides by 5.5 to find the value of h: \(h = \frac{33 \cdot 4}{5.5}\)

Evaluate h and round to the nearest foot

Calculate the value of h: \(h = \frac{132}{5.5} = 24\) Since the problem asks for the height of the tree rounded to the nearest foot, our final answer is \(h = 24\) feet. The correct choice is (A) 24 feet.

Key concepts

Understanding Proportions

Proportions are equations that express two ratios as being equal. In the tree problem, we compare the height of the tree to the length of its shadow. Similarly, we do this for the mailbox and its shadow. The key idea is that these two ratios should be equal because they are cast under the same conditions.
A proportion can be written as two fractions set equal to each other. For example, the height of the tree over its shadow length is compared to the mailbox's height over its shadow length. Formally, this is represented as:

• The height of the tree \( h \) over its shadow: \( \frac{h}{33} \)
• The height of the mailbox over its shadow: \( \frac{4}{5.5} \)

Setting these equal gives us the proportion \( \frac{h}{33} = \frac{4}{5.5} \). This proportion helps us to find the unknown height of the tree using known lengths of shadows and the mailbox height.

Effective Problem-solving Techniques

Solving problems often involves breaking them into smaller, manageable parts. The tree and mailbox shadow problem is an excellent example where establishing a clear relationship between known and unknown variables is crucial. Start by identifying what you know (like the lengths of the shadows and the height of the mailbox) and what you need to find, which is the height of the tree.
By setting up a proportion, you create a simple equation that relates these knowns to the unknown. After establishing the proportion, solving for the unknown involves straightforward algebraic manipulation:

• Cross-multiply to eliminate fractions: \( 5.5h = 33 \cdot 4 \)
• Divide to isolate the unknown variable: \( h = \frac{132}{5.5} \)

Simplifying this will yield the tree's height. Practice by solving similar problems to gain confidence and develop efficient methods for tackling proportional reasoning.

Basic Geometry in Everyday Contexts

Geometry isn't just about shapes; it's about understanding sizes, relationships, and properties of things in space. In our example, we're dealing with lengths and heights. Even something as simple as shadows can demonstrate geometric concepts.
Shadows are great geometric tools because they are predictable based on light angles and object heights. By comparing shadows, we can use proportions to handle problems involving sizes and distances without direct measurement. In doing so, you dive into the practical application of geometry where shapes (in this case, the silhouette or shadow) can provide valuable real-world insights.

Mastering Measurement Conversions

Measurement conversions are key in ensuring measurements are usable and understandable. Although the tree problem does not require conversion between different units, understanding conversion can be crucial in different scenarios.
Conversions help maintain consistency, especially when measurements are in diverse units like meters, feet, or inches. The ability to convert measurements ensures that problems remain mathematically accurate and relatable. Remember these steps to master conversions:

• Identify the units you have and the units you want to convert to
• Use appropriate conversion factors (like \(1\) foot = \(12\) inches or \(1\) meter = \(3.28084\) feet)
• Multiply or divide to perform the conversion

This exercise ensured uniformity of units by maintaining measurements in feet, simplifying the calculation and comparison processes. Always double-check your final units after any calculation to make sure they are what the problem requires.