Problem 64 - A Graphical Approach to Precalculus with Limits

Question

A formula for calculating the distance one can see from an airplane to the horizon on a clear day is given by \(f(x)=1.22 x^{0.5},\) where \(x\) is the altitude of the plane in feet and \(f(x)\) is in miles. If a plane is flying at \(30,000\) feet, how far can the pilot see?

Step-by-Step Solution

Answer

The pilot can see approximately 211.67 miles to the horizon.

Step 1: Identify the Variables

First, identify what each variable in the formula represents. Here, \(x\) is the altitude in feet, and \(f(x)\) is the distance to the horizon in miles.

Step 2: Substitute Altitude into the Formula

The formula for calculating the distance visible from an airplane to the horizon is \(f(x)=1.22x^{0.5}\). Substitute \(x=30,000\) into the equation.

Step 3: Calculate the Square Root

Compute the square root of 30,000. This is \(x^{0.5}=\sqrt{30,000}\).

Step 4: Multiply by 1.22

Plug the value obtained from the square root calculation into the formula and multiply by 1.22 to find \(f(x)\).

Step 5: Solve the Expression

Multiply \(1.22\) by \(\sqrt{30,000}\) to get the final result. This tells us the distance to the horizon the pilot can see.

Key Concepts

Square Root Calculation

Calculating the square root is essential when the equation involves an exponent of 0.5. The square root of a number is a value that, when multiplied by itself, returns the original number. For example, the square root of 9 is 3, because when you multiply 3 by itself (3 x 3), you get 9. In mathematical notation, \( \sqrt{30,000} \) represents the square root of 30,000.

If you are calculating by hand, consider breaking the number into smaller parts. This number is large, but you might use estimation or a calculator for a more precise figure.

• Estimate a nearby perfect square. - For instance, 30,000 is near to 30,625, which is 175 squared.
• Use a calculator if exact precision is needed.

Understanding square roots is vital for more complex calculations such as this one or in various other mathematical principles.

Distance Formula

The distance formula in our context determines how far a pilot can see to the horizon, based on the plane's altitude. This formula helps convert the vertical height above sea level to a horizontal distance to the horizon. It's calculated using the equation \( f(x) = 1.22 \times x^{0.5} \), where \( x \) is the altitude in feet and \( f(x) \) gives the visible distance in miles.

In simpler terms, when you increase the altitude \( x \), you increase the visible distance \( f(x) \). Conceptually, when you're higher up, you have fewer obstacles obstructing your view, so you can see further.

• Step 1: Determine altitude (in feet)
• Step 2: Calculate the square root of the altitude
• Step 3: Multiply result with 1.22 to get the visible horizon distance in miles

This formula provides a quick way to gauge how far you can see from high altitudes without needing advanced mathematical tools.

Altitude and Horizon Distance

Altitude refers to the height above a reference point, like sea level. In aviation, altitude plays a key role in many calculations, including how far a pilot can see the horizon.

As you climb higher, the line of sight extends over greater distances due to the Earth's curvature. The higher you are, the further away the horizon appears. Hence, altitude and visibility are directly related.

• Higher altitude means clearer view toward the horizon.
• The Earth's curvature limits how far is visible at lower altitudes.

The formula \( f(x)=1.22x^{0.5} \), as explained, lets us quantify this relationship, showing us precisely how the visible horizon distance shifts with changes in altitude. Understanding how these concepts interrelate can enhance comprehension of pilot visibility and navigation, especially during flight planning.