Problem 80 - A Graphical Approach to College Algebra

Question

Solve each problem. A formula for calculating the distance \(d\) one can see from an airplane to the horizon on a clear day is $$d=1.22 \sqrt{x}$$ where \(x\) is the altitude of the plane in feet and \(d\) is given in miles. How far can one see to the horizon in a plane flying at each altitude? Give answers to the nearest mile. (a) \(15,000\) feet? (b) \(20,000\) feet?

Step-by-Step Solution

Answer

(a) 149 miles (b) 173 miles

Step 1: Understand the Formula

The formula provided is \( d = 1.22 \sqrt{x} \), which calculates the distance to the horizon \( d \) in miles based on the altitude of the airplane \( x \) in feet. Our task is to compute \( d \) for given altitudes.

Step 2: Plug in the Altitude for Case (a)

For part (a), the altitude \( x \) is given as 15,000 feet. Substitute 15,000 into the formula: \( d = 1.22 \sqrt{15,000} \).

Step 3: Compute the Square Root

Calculate the square root of 15,000: \( \sqrt{15,000} \approx 122.47 \).

Step 4: Multiply by the Constant

Multiply the square root by 1.22 to find the distance: \( d = 1.22 \times 122.47 \approx 149.41 \).

Step 5: Round to the Nearest Mile for (a)

Round 149.41 to the nearest mile, resulting in \( d \approx 149 \) miles for an altitude of 15,000 feet.

Step 6: Plug in the Altitude for Case (b)

For part (b), the altitude \( x \) is 20,000 feet. Substitute 20,000 into the formula: \( d = 1.22 \sqrt{20,000} \).

Step 7: Compute the Square Root for (b)

Calculate the square root of 20,000: \( \sqrt{20,000} \approx 141.42 \).

Step 8: Multiply by the Constant for (b)

Multiply the square root by 1.22: \( d = 1.22 \times 141.42 \approx 172.53 \).

Step 9: Round to the Nearest Mile for (b)

Round 172.53 to the nearest mile, resulting in \( d \approx 173 \) miles for an altitude of 20,000 feet.

Key Concepts

Understanding Altitude in the Distance Formula

Altitude is a crucial concept in understanding how far an observer can see to the horizon from an airplane. In the context of this exercise, altitude refers to the height of the airplane above the ground, measured in feet.
The greater the altitude, the further one can typically see due to the earth's curvature.
Here, altitude is represented by the variable \( x \) in the formula \( d = 1.22 \sqrt{x} \) for calculating the distance \( d \) one can see from the airplane to the horizon. It's important to accurately determine the altitude of the plane in feet, as this value plays a direct role in the calculation of distance to the horizon. By ensuring that the altitude is correctly identified, the formula will provide a valid distance that is based on geometrical and optical principles. Altitude can vary widely for different flights and conditions, affecting the distance; our exercise examines specific altitudes (like 15,000 and 20,000 feet) to demonstrate how distance changes with altitude in the formula.
It's fascinating how the higher the plane, the further passengers can see, offering a broader view of the landscape and horizon below.

Square Root Calculation in the Formula

The square root calculation is a key mathematical operation used to evaluate the distance formula \(d = 1.22 \sqrt{x}\).
The symbol \(\sqrt{ }\) represents the square root, which is a value that, when multiplied by itself, gives the original number.
For example, if we're finding the square root of \(15,000\), we're looking for a number that, when squared, equals \(15,000\).When plugging the altitude \( x \) values of \(15,000\) or \(20,000\) into the formula, we need to find \(\sqrt{15,000} \approx 122.47\) and \(\sqrt{20,000} \approx 141.42\), respectively. These computations are crucial steps, as they prepare us to apply the remainder of the formula. Square root calculations can be performed using a calculator for precision; a rough mental approximation is also useful in some contexts but might not always be as accurate for larger numbers like those in our exercise.Understanding the square root helps in dissecting mathematical formulas, making it a valuable skill not just for this exercise, but for many other problems you'll encounter in algebra, geometry, and beyond.

Mathematical Rounding to the Nearest Mile

Rounding is a mathematical technique used to simplify numbers, particularly in instances where precision up to several decimal places is unnecessary or impractical.
In this exercise, we're asked to round the calculated distances to the nearest mile.
This involves analyzing the decimal part of the number and rounding it based on standard rules. How to Round:

• If the first decimal place is 5 or greater, round up.
• If the first decimal place is less than 5, round down.

For instance, the computed distance at 15,000 feet was \(149.41\) miles.
The first decimal is 4, which means we round down to 149 miles.
Similarly, for 20,000 feet, \(172.53\) rounds up to 173 miles because the decimal part is 5 or more.This process ensures clarity and usability for results, especially in practical fields that rely on straightforward interpretations. By mastering rounding, you convey quantitative results more effectively while adhering to requirements or limits often found in real-world applications. Using simple rounding, the results become easier to understand at a glance, which is particularly helpful in navigation and other practical computations.