The lifting force, F, exerted on an airplane wing varies jointly as the area, A, of the wing's surface and the square of the plane's velocity, v. The lift of a wing with an area of 220
 square feet is 24,700
 pounds when the plane is going at 160
 miles per hour. Find the lifting force if the speed is 110
 miles per hour. Round your answer to the nearest integer if necessary.

The lifting force \( F \) on an airplane wing varies jointly with the area \( A \) of the wing's surface and the square of the velocity \( v \). This relationship can be expressed as:

\[
F = k \cdot A \cdot v^2
\]

Where:
- \( F \) is the lifting force,
- \( A \) is the area of the wing,
- \( v \) is the velocity,
- \( k \) is a constant of proportionality.

### Step 1: Find the constant \( k \)

We are given:
- \( A = 220 \, \text{square feet} \),
- \( v = 160 \, \text{miles per hour} \),
- \( F = 24,700 \, \text{pounds} \).

Substitute these values into the equation:

\[
24,700 = k \cdot 220 \cdot (160)^2
\]

Now solve for \( k \):

\[
24,700 = k \cdot 220 \cdot 25,600
\]

\[
24,700 = k \cdot 5,632,000
\]

\[
k = \frac{24,700}{5,632,000}
\]

\[
k \approx 0.0043865
\]

### Step 2: Calculate the lifting force at \( v = 110 \) mph

Now that we have the value of \( k \), we can calculate the lifting force when the speed is 110 miles per hour. Using the same formula:

\[
F = 0.0043865 \cdot 220 \cdot (110)^2
\]

\[
F = 0.0043865 \cdot 220 \cdot 12,100
\]

\[
F \approx 0.0043865 \cdot 2,662,000
\]

\[
F \approx 11,676.04 \, \text{pounds}
\]

### Final Answer:
The lifting force is approximately **11,676 pounds** when the speed is 110 miles per hour.

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:
1. How does the lifting force change if the velocity is doubled?
2. What would be the lifting force if the area of the wing increased by 50%?
3. How would the lift change if the velocity is reduced to 80 miles per hour?
4. What happens to the lifting force if both the area and velocity are halved?
5. Can we apply this equation to calculate lift for wings of different shapes?

**Tip**: When solving joint variation problems, always identify the constant of proportionality \( k \) first, as it simplifies all future calculations.

The lifting force, F, exerted on an airplane wing varies jointly as the area, A, of the wing's surface and the square of the plane's velocity, v. The lift of a wing with an area of 220 square feet is 24,700 pounds when the plane is going at 160 miles per hour. Find the lifting force if the speed is 110 miles per hour. Round your answer to the nearest integer if necessary.

Learn how to calculate the lifting force on an airplane wing using the formula F = k * A * v^2. Given a wing area of 220 square feet and initial lift at 160 mph, find the lift at 110 mph. Step-by-step solution suitable for Grades 9-12.

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