Problem 32 - Algebra and Trigonometry

Question

Aerodynamic Lift The lift \(L\) on an airplane wing at take- off varies jointly as the square of the speed \(s\) of the plane and the area \(A\) of its wings. A plane with a wing area of 500 \(\mathrm{ft}^{2}\) traveling at 50 \(\mathrm{mi} / \mathrm{h}\) experiences a lift of 1700 \(\mathrm{ib}\) . How much lift would a plane with a wing area of 600 \(\mathrm{ft}^{2}\) traveling at 40 milh experience?

Step-by-Step Solution

Answer

The plane would experience a lift of 1305.6 lb.

Step 1: Determine the joint variation equation

The lift \(L\) varies jointly with the square of the speed \(s\) and the area \(A\). This can be expressed as \(L = k \cdot s^2 \cdot A\), where \(k\) is the constant of proportionality.

Step 2: Calculate constant of proportionality

Given: Lift \(L = 1700\) lb, speed \(s = 50\) mi/h, and area \(A = 500\) ft². Substituting these values into the equation: \(1700 = k \cdot (50)^2 \cdot 500\).Simplify to find \(k\): \(1700 = k \cdot 2500 \cdot 500\), \(k = \frac{1700}{2500 \times 500}\), \(k = \frac{1700}{1250000}\), \(k = 0.00136\).

Step 3: Determine new lift with different parameters

Use the calculated \(k = 0.00136\) to find the new lift. Given: area \(A = 600\) ft², speed \(s = 40\) mi/h.Substitute into the equation: \(L = 0.00136 \cdot (40)^2 \cdot 600\).Calculate: \(L = 0.00136 \cdot 1600 \cdot 600\), \(L = 0.00136 \times 960000\), \(L = 1305.6\).

Key Concepts

Aerodynamic Lift

Aerodynamic lift is a crucial concept in aviation that explains how planes achieve and maintain flight. This lift is the force that opposes gravity and enables an airplane to ascend from the ground. The lift force is created by the wings and is impacted by the speed of the airplane and the design of the wing, specifically the area.

In our example, the lift force (\(L\)) is described as varying jointly with both the speed (\(s\)) of the airplane squared and the wing area (\(A\)). This means:

• Increasing the speed of the airplane results in more lift.
• A larger wing area can also enhance lift.

Thus, both factors are critically important in determining how much lift is generated during take-off.

Proportionality Constant

The proportionality constant (\(k\)) is a key aspect of understanding joint variation problems. It acts as a fixed number that ties together variables in a mathematical relationship to let us predict different outcomes. In joint variation, this constant reveals how changes in speed and wing area affect lift.

To determine the constant, we use known values for lift, speed, and wing area in the equation: \[ L = k imes s^2 imes A \]By substituting the values (e.g., \(L = 1700\) lb, \(s = 50\) mi/h, \(A = 500\) ft²), we solve for \(k\):

• \(k = \frac{1700}{2500 \times 500}\)
• \(k = \frac{1700}{1250000}\)
• \(k = 0.00136\)

This constant then applies to different scenarios with different speeds and wing areas, allowing us to calculate new lift values.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and predict real-world phenomena. In studying aerodynamic lift, mathematical models help assess how changes in variables like speed and wing area influence flight performance.

The joint variation equation is a mathematical model showing the relationship between speed, wing area, and lift. This powerful tool can:

• Help engineers design effective wing shapes for optimal lift.
• Allow pilots to understand the essentials behind controlling an aircraft.
• Inform safety protocols and efficiency measures in aviation.

By using accurate values and calculations, these models ensure reliable and safe aircraft design and operation.