Let's denote the usual speed of the plane as $v$ km/hr.

Step 1: Calculate the time it would take to travel 1600 km at the usual speed

The time taken to travel 1600 km at the usual speed $v$ is given by: $\text{Time at usual speed} = \frac{1600}{v} \text{ hours}$

Step 2: Determine the time the plane actually had due to the delay

Since the plane left 40 minutes late, it had 40 minutes less to reach the destination on time. 40 minutes is equivalent to $\frac{40}{60} = \frac{2}{3}$ hours.

So, the time the plane actually had is: $\text{Available time} = \frac{1600}{v} - \frac{2}{3} \text{ hours}$

Step 3: Set up the equation with the increased speed

When the plane increases its speed by 400 km/hr, its new speed becomes $v + 400$ km/hr. The time taken to travel 1600 km at this increased speed is: $\text{Time at increased speed} = \frac{1600}{v + 400} \text{ hours}$

Since the plane must cover the distance in the reduced time: $\frac{1600}{v + 400} = \frac{1600}{v} - \frac{2}{3}$

Step 4: Solve the equation

Multiply both sides of the equation by $(v + 400) \times v$ to clear the denominators: $1600v = 1600(v + 400) - \frac{2}{3} v(v + 400)$

Expanding and simplifying: $1600v = 1600v + 640000 - \frac{2}{3}v^2 - \frac{800}{3}v$ $0 = 640000 - \frac{2}{3}v^2 - \frac{800}{3}v$

Multiply through by 3 to eliminate the fractions: $0 = 1920000 - 2v^2 - 800v$ $2v^2 + 800v - 1920000 = 0$

Step 5: Solve the quadratic equation

Divide by 2: $v^2 + 400v - 960000 = 0$

Use the quadratic formula $v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 400$, and $c = -960000$: $v = \frac{-400 \pm \sqrt{400^2 + 4 \times 960000}}{2}$ $v = \frac{-400 \pm \sqrt{160000 + 3840000}}{2}$ $v = \frac{-400 \pm \sqrt{4000000}}{2}$ $v = \frac{-400 \pm 2000}{2}$

This gives us two possible solutions: $v = \frac{1600}{2} = 800 \text{ km/hr} \quad \text{or} \quad v = \frac{-2400}{2} = -1200 \text{ km/hr}$

Since speed cannot be negative, the usual speed $v$ is: $\boxed{800 \text{ km/hr}}$

Summary

The usual speed of the plane is 800 km/hr.

Would you like more details or have any questions? Here are some related questions:

1. How do you apply the quadratic formula to solve equations?
2. What are the properties of quadratic equations?
3. How does increasing speed affect travel time?
4. What are the practical applications of such problems in aviation?
5. How can one convert minutes to hours in time calculations?

Tip: Always check the units and convert them to match for consistency when solving problems involving time, speed, and distance.