Problem 11 - Prealgebra

Question

Write a proportion. Then determine the missing measure. RIDES Suppose a roller coaster casts a shadow of 31.5 feet. At the same time, a nearby Ferris wheel casts a 19 -foot shadow. A sign says the roller coaster is 126 feet tall. How tall is the Ferris wheel?

Step-by-Step Solution

Answer

The Ferris wheel is 76 feet tall.

Step 1: Identify Proportional Relationships

The problem states that both a roller coaster and a Ferris wheel cast shadows. We can assume the sun casts light at the same angle on both structures, so the ratio of the heights to their shadows is equal for both. This gives us the proportion: \( \frac{\text{Height of Roller Coaster}}{\text{Shadow of Roller Coaster}} = \frac{\text{Height of Ferris Wheel}}{\text{Shadow of Ferris Wheel}} \).

Step 2: Set Up the Proportion

Using the values given: the roller coaster is 126 feet tall and casts a shadow of 31.5 feet, while the Ferris wheel casts a shadow of 19 feet. Substituting these values into the proportion gives: \( \frac{126}{31.5} = \frac{x}{19} \), where \(x\) is the height of the Ferris wheel.

Step 3: Solve for the Missing Measure

To solve the proportion, we cross-multiply: \(126 \times 19 = 31.5 \times x\). Simplifying the left side, we get 2394. Solving for \(x\) gives \( x = \frac{2394}{31.5} \).

Step 4: Calculate the Height of the Ferris Wheel

Now, compute \( x = \frac{2394}{31.5} \) to find the height of the Ferris wheel. Performing this division, \( x = 76 \). Therefore, the height of the Ferris wheel is \(76\) feet.

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful tool used to solve proportions. Proportions are equations that state two ratios are equal. When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), you can use cross-multiplication to solve it by multiplying across the equals sign diagonally. This gives you the equation \( a \times d = b \times c \).
This method simplifies solving for an unknown variable in proportions because it turns the equation into a straightforward multiplication problem.
For example, in our exercise, we have the proportion \( \frac{126}{31.5} = \frac{x}{19} \). Cross-multiplying gives us \( 126 \times 19 = 31.5 \times x \). This step helps us isolate the variable, making it easier to find the unknown value through basic division.

This technique is especially useful in "shadow problems" like our exercise, where you can compare two similar situations that share a constant ratio, such as shadow lengths and object heights.

Shadow Problems

Shadow problems are a classic example of how geometry and proportions come together in real life. They arise when two objects of different heights cast shadows due to the same light source, creating proportional relationships.

In these types of problems, the position of the sun creates similar triangles between the object and its shadow. Because the triangles are similar, their corresponding sides are proportional. This understanding helps us set up equations to solve for unknown measurements like the height of an object.

To tackle shadow problems, follow these steps:

• Identify objects and their corresponding shadow lengths.
• Set up a proportion using the known lengths and heights.
• Use cross-multiplication to solve for the unknown.

Understanding shadow problems enhances our grasp of applying geometric concepts to practical scenarios, like determining the height of a Ferris wheel given its shadow.

Ratios

Ratios are fundamental to understanding proportions. A ratio compares two quantities and is often represented as a fraction. For example, the ratio of the height to the shadow of the roller coaster in our earlier exercise is \( \frac{126}{31.5} \).

Ratios are everywhere. They help us compare sizes, speeds, and other quantities. They are essential for maintaining consistency and scale. In our exercise, ratios are used to make a direct comparison between the roller coaster and the Ferris wheel, showing how their heights relate to their shadow lengths.

Working with ratios involves:

• Recognizing proportional relationships.
• Setting up equations to express these relationships.
• Solving equations through cross-multiplication.

Strengthening your skills with ratios prepares you for more complex problem-solving in areas such as engineering, architecture, and physics.

Geometry in Real Life

Geometry is much more than just a subject in school; it's a tool we use to understand the world. "Geometry in real life" involves applying geometric principles to practical situations, such as building design, navigation, and even solving our shadow problem.

The principles of geometry allow us to calculate, measure, and understand the spatial relationships between different objects. When you apply geometry to real-world scenarios, it helps bridge the gap between theoretical concepts and practical applications.

In the context of our exercise, geometry allows us to understand how light creates shadows and how those shadows relate back to the objects casting them. This principle is essential for architects and designers who use shadows to plan for structural positioning relative to the sun.

Using geometry in real-life applications teaches us to see math as a key component in solving everyday challenges, encouraging a deeper appreciation for the subject beyond the classroom.