Problem 81 - Intermediate Algebra

Question

Use similar triangles to solve each problem. Flight Paths. \(\quad\) An airplane ascends 150 feet as it flies a horizontal distance of \(1,000\) feet. How much altitude will it gain as it flies a horizontal distance of 1 mile? (Hint: \(5,280 \text { feet }=1 \text { mile. })\)

Step-by-Step Solution

Answer

The airplane gains 792 feet of altitude over 1 mile.

Step 1: Identify the Ratio of Ascension

Given that the airplane ascends 150 feet for every 1,000 feet of horizontal distance, set up a ratio to describe this relationship.\[ \text{Ratio of ascent} = \frac{150}{1000} \]

Step 2: Convert Mile to Feet

Convert the given horizontal distance from miles to feet to match the units of the ascent rate. A mile equals 5,280 feet.

Step 3: Set Up the Proportion

Using the similar triangles principle, set up a proportion with the ascent ratio and the new horizontal distance.\[ \frac{150}{1000} = \frac{x}{5280} \]where \(x\) is the altitude gained over 5,280 feet.

Step 4: Solve for the Unknown Altitude

Cross-multiply and solve for \(x\) to find the amount of altitude gained over 5,280 feet.\[ 150 \times 5280 = 1000 \times x \]\[ 792000 = 1000x \]\[ x = \frac{792000}{1000} \]\[ x = 792 \]

Step 5: Conclusion

The airplane will gain 792 feet of altitude after flying a horizontal distance of 1 mile.

Key Concepts

Proportion

In mathematics, a proportion is a statement that two ratios are equal. This concept is widely used when comparing different quantities, enabling us to determine one quantity when another is known. For example, in the flight path problem, we have two connected quantities: ascension and horizontal distance. We express these using a proportion. The airplane ascends 150 feet for every 1,000 feet of horizontal travel. To find out how much it will ascend over a horizontal mile, we set up a proportion based on these values:

• Original ascent ratio: \( \frac{150}{1000} \)
• New horizontal distance: 5,280 feet (1 mile)

Setting up the equation: \( \frac{150}{1000} = \frac{x}{5280} \), where \(x\) represents the unknown altitude gained over one mile.

Ratio

Ratios are used to compare two quantities by division. This can be applied to virtually everything: lengths, weights, times, and more. In the context of similar triangles, ratios help us understand relationships between corresponding sides of the triangles. In our airplane example, the ratio of ascension per unit of horizontal distance (150 feet ascended per 1,000 feet traveled) is crucial for calculating the ascent over a different horizontal distance.

• Ratios are expressed as fractions or with a colon, like 3:2 or \( \frac{3}{2} \).
• They provide a means of scaling quantities up or down proportionally.

By utilizing the initial ratio of \( \frac{150}{1000} \), we can apply it to solve for new situations, like the plane's ascent across a mile.

Geometry

Geometry involves understanding the properties and relationships of points, lines, surfaces, and solids. Similar triangles are a fundamental concept in this field.

• Similar triangles have the same shape but not necessarily the same size.
• Corresponding angles are equal, and corresponding sides are proportional.

In the flight path problem, the concept of similar triangles allows us to form a reasonable comparison between the initial short ascent path and the extended mile-long path. Although we don't visually draw the triangles, the idea of using the same angles (and therefore proportional sides) is implicit in the equation. Understanding these properties lets us cleverly bypass direct measurement, relying instead on ratios and proportions.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It provides tools for solving equations and finding unknown values. In the context of our airplane problem, algebra allows us to solve for the unknown in the proportion \( \frac{150}{1000} = \frac{x}{5280} \).Here's how we solve this step-by-step:

• Cross-multiply to eliminate the fractions: \( 150 \times 5280 = 1000 \times x \)
• Calculate the product: \( 792000 = 1000x \)
• Divide by 1000 to isolate \( x \): \( x = \frac{792000}{1000} \)

The solution, \( x = 792 \), tells us how many feet the airplane ascends over a mile. This highlights algebra's role in converting mathematical relationships into actionable answers.