Problem 103 - Algebra for College Students

Question

Solve each problem. The formula \(1211.1 L=C A^{2} S\) is used to determine the approach speed for landing an aircraft, where \(L\) is the gross weight of the aircraft in pounds, \(C\) is the coefficient of lift, \(S\) is the surface area of the wings in square feet ( \(\mathrm{ft}^{2}\) ), and \(A\) is approach speed in feet per second. Find \(A\) for the Piper Cheyenne, which has a gross weight of 8700 lb, a coefficient of lift of \(2.81,\) and wing surface area of \(200 \mathrm{ft}^{2}\).

Step-by-Step Solution

Answer

The approach speed \(A\) is approximately 137 \(\text{ft/s}\).

Step 1: Identify the given values

Identify the given values for the Piper Cheyenne: \(L = 8700\) lbs (gross weight of the aircraft)\(C = 2.81\) (coefficient of lift)\(S = 200\) \(\text{ft}^2\) (surface area of the wings)

Step 2: Write down the formula

Write the formula given in the problem: \[1211.1L = C A^2 S\]

Step 3: Substitute the values into the formula

Substitute the given values of \(L\), \(C\), and \(S\) into the formula: \[1211.1 \times 8700 = 2.81 \times A^2 \times 200\]

Step 4: Simplify the equation

Solve for \(A^2\) by simplifying the equation: \[1211.1 \times 8700 = 2.81 \times A^2 \times 200\] \[10,534,170 = 562A^2\]

Step 5: Isolate \(A^2\)

Divide both sides of the equation by 562 to isolate \(A^2\):\[A^2 = \frac{10,534,170}{562}\]\[A^2 = 18,747\]

Step 6: Solve for \(A\)

Find the square root of both sides to solve for \(A\):\[A = \sqrt{18,747}\]\[A \approx 137\] \(\text{ft/s}\)

Key Concepts

Algebraic Equation

In mathematics, an algebraic equation is an expression that sets two algebraic expressions equal to one another. Algebraic equations involve variables and constants. In the context of calculating the aircraft landing speed, we start with the given formula for approach speed:
\[1211.1L = CA^2S\]
The variables here represent:

• L: Gross weight of the aircraft (lbs)
• C: Coefficient of lift
• S: Surface area of the wings (\text{ft}^2)
• A: Approach speed (ft/s)

To solve for the approach speed, we need to manipulate this algebraic equation systematically.

Solving Equations

Solving algebraic equations involves finding the value of the variable that makes the equation true. The steps typically include:

• Identifying given values and substituting them into the equation.
• Simplifying the equation step-by-step until the variable is isolated.

Here, for the Piper Cheyenne, we know the given values: \(L = 8700\) lbs, \(C = 2.81\), and \(S = 200 \text{ft}^2\).
By substituting these values, the equation becomes:
\[1211.1 \times 8700 = 2.81 \times A^2 \times 200\]
Next, we simplify the right-hand side and isolate \(A^2\):
\[10534170 = 562A^2\]
Finally, we divide both sides by 562 to solve for \(A^2\).

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, if \(x^2 = y\), then \(x = \sqrt{y}\).
To find the approach speed \(A\) for the aircraft, we reached the equation:
\[A^2 = 18747\]
Taking the square root of both sides, we get:
\[A = \sqrt{18747}\]
\(A\) approximately equals \(137\) ft/sec. This reveals the approach speed of the aircraft. In practical terms, taking the square root helps us solve for the value of the speed from an equation with a squared term. Understanding how to work with square roots simplifies the process of solving complex equations involving squared variables.