# Solve Derivative Problem: Plane Descent from h to (0,0)

• kristo
In summary, the conversation discusses a problem involving a plane's descent and landing path. The focus is on finding a, b, c, and d to ensure a smooth landing, as well as determining the values of dx/dt, dy/dt, and d2y/dt2 at specific points. The solution involves setting certain conditions for the derivative and second derivative, and using the chain rule to find dy/dt.
kristo

## Homework Statement

A plane starts its descent from height y=h at x =-L to land at (0,0). Choose a, b, c, d so its landing path y=ax^3 + bx^2 + cx + d is smooth.With dx/dt=V=constant, find dy/dt and d2y/dt2 at x=0 and x =-L. (To keep d2y/dt2 small, a coast-to-coast plane starts down L>100 miles from the airport.)

## The Attempt at a Solution

For a smooth landing the derivative must be negative and 0 at x=0.
y'=3ax^2+2bx+c < 0
y'(0)=0
c=0
d must be 0 because y(0)=0.
second derivative must be > 0.
y''=6ax+2b >0

And that's as far as I get. I'd appreciate any tips and can someone explain to me how dx/dt can be constant? And how do I find dy/dt?

Merry Christmas!

kristo said:
… can someone explain to me how dx/dt can be constant? And how do I find dy/dt?

Hi kristo!

Think of the curve as a hillside … the driver can drive at whatever speed he likes.

And to find dy/dt, use the chain rule.

## 1. What is the formula for finding the derivative of a plane's descent from a given height to the origin (0,0)?

The formula for finding the derivative of a plane's descent is d = (h-0)/(t-0), where d represents the distance, h represents the initial height, and t represents the time taken to descend to the origin (0,0).

## 2. How do I solve for the initial height if the distance and time are known?

To solve for the initial height, you can rearrange the formula to be h = dt. Plug in the given values for distance and time, and you will get the initial height.

## 3. Can the derivative of a plane's descent be negative?

Yes, the derivative of a plane's descent can be negative. A negative value indicates that the plane is descending, while a positive value indicates that the plane is ascending.

## 4. How does air resistance affect the derivative of a plane's descent?

Air resistance can affect the derivative of a plane's descent by slowing down the rate of descent. This would result in a smaller derivative value, meaning the plane is not descending as quickly.

## 5. Is the derivative of a plane's descent affected by the weight of the plane?

Yes, the derivative of a plane's descent is affected by the weight of the plane. A heavier plane will require more force to descend, resulting in a larger derivative value.

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