Problem 38 - Precalculus Mathematics for Calculus

Question

The lift \(L\) on an airplane wing at takeoff 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 mi/h experiences a lift of 1700 lb. How much lift would a plane with a wing area of \(600 \mathrm{ft}^{2}\) traveling at \(40 \mathrm{mi} / \mathrm{h}\) experience?

Step-by-Step Solution

Answer

The lift for the new conditions is 1306 lb.

Step 1: Write the Joint Variation Formula

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

Step 2: Find the Constant of Proportionality \( k \)

Using the given information, substitute the known values into the equation. You have \( L = 1700 \) lb, \( s = 50 \) mi/h, and \( A = 500 \) ft\(^2\). Thus, \( 1700 = k \cdot (50)^2 \cdot 500 \). Solving for \( k \), we find:\[k = \frac{1700}{50^2 \cdot 500} = \frac{1700}{1250000} = 0.00136.\]

Step 3: Calculate the New Lift \( L \) for Different Speed and Area

Now that we have the constant \( k = 0.00136 \), use it to find the new lift. Substitute \( s = 40 \) mi/h and \( A = 600 \) ft\(^2\) into the equation: \[L = 0.00136 \cdot (40)^2 \cdot 600.\]Calculate \( L \) to find the lift:\[L = 0.00136 \cdot 1600 \cdot 600 = 0.00136 \cdot 960000 = 1305.6.\]

Step 4: Conclusion

The new lift, when the plane travels at 40 mi/h with a wing area of 600 ft², is approximately 1306 lb (rounded to the nearest whole number).

Key Concepts

Constant of Proportionality

Joint variation describes a situation where a variable depends on multiple other variables in a specific manner. In our scenario, lift depends on the square of speed and wing area.
The constant of proportionality, denoted by \( k \), acts as a bridge between these variables and their effect on lift.

• Think of it as a fixed element that maintains the relationship between speed, area, and lift.
• Without \( k \), you can't calculate the lift accurately for different speeds and wing areas.

By working out \( k \) using known values, you establish a standard for all calculations. Here, you plug in the known lift, speed, and area values to find \( k = 0.00136 \).
This constant helps translate speeds and areas into quantifiable lift values for various scenarios.

Lift Calculation

In the world of aerodynamics, calculating the lift is crucial. Lift represents the upward force keeping planes airborne.
To compute this when speed or wing area changes, the formula allows us to adjust accordingly.

• The formula, \( L = k \cdot s^2 \cdot A \), shows how changing both speed and wing area impacts the lift.
• By substituting different speeds and areas, you predict how these changes affect the plane's ability to lift.

In our example, calculating lift for a different speed (40 mi/h) and area (600 ft²) involved plugging values into the formula and solving.
The result was a lift of approximately 1306 lb, showing the vital role lift calculations play in aeronautics for ensuring flights are safe and efficient.

Aerodynamics

Understanding aerodynamics is pivotal in grasping how planes fly. It's the study of air motion, particularly as it interacts with solid objects like airplane wings.
In aerodynamics, lift is a result of air pressure differences created by wing shape and speed.

• A wing's shape is crucial as it dictates how air travels over and under it.
• Higher speeds increase air flow, impacting lift directly.

The relationship between wing area, speed, and lift illustrates aerodynamics principles at work.
Knowing how to compute lift using these concepts helps express fundamental aerodynamic theories practically, emphasizing its importance in real-world flight scenarios.