https://file.aiquickdraw.com/mathbot/file/1729240898114cldvty2n.jpg

The problem presented asks for the minimum runway length for a small plane, based on the given position function \( x(t) = 1.3t^2 \), where \( t \) is time in seconds, and the minimum takeoff speed of 33.8 m/s.

### To solve this:
1. The given position function is \( x(t) = 1.3t^2 \).
2. The velocity \( v(t) \) is the derivative of the position function with respect to time:
   \[
   v(t) = \frac{dx}{dt} = 2 \times 1.3t = 2.6t
   \]
3. We are given the minimum takeoff speed, \( v = 33.8 \, \text{m/s} \). To find the time at which the plane reaches this speed, we set the velocity equal to 33.8 m/s:
   \[
   33.8 = 2.6t \quad \Rightarrow \quad t = \frac{33.8}{2.6} \approx 13 \, \text{seconds}
   \]
4. Now, substitute \( t = 13 \) seconds into the position function to find the minimum runway length:
   \[
   x(13) = 1.3(13^2) = 1.3 \times 169 = 219.7 \, \text{meters}
   \]

Therefore, the minimum required runway length is **219.7 meters**.

---

Do you want more details or have any questions?  
Here are some related questions:

1. How do we derive velocity from the position function?
2. What is the physical significance of the time we calculated?
3. How would the required runway length change if the takeoff speed were higher?
4. What would the function look like if the acceleration were not constant?
5. Can we calculate the acceleration of the plane based on this problem?

**Tip**: Always ensure to differentiate correctly when dealing with velocity and acceleration in physics problems based on position functions.

The position from its starting point of a small plane preparing for takeoff is given by x(t) = 1.3t^2 meters (t is measured in seconds). If the minimum takeoff speed of the plane is 33.8 m/s, what is the minimum required runway length for this type of plane?

This problem involves calculating the minimum runway length for a plane using the position function x(t) = 1.3t^2 and a minimum takeoff speed of 33.8 m/s. The solution covers key kinematics concepts, including derivatives and velocity calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation