An airplane flying at 115 m/s due east makes a gradual turn following a circular path to fly south. The turn takes 15 seconds to complete. What is the radius of the curve that the plane follows in making the turn

To solve this problem, we must note that the airplane does not make a full circle. Instead, it turns from flying east to flying south. Check the attachment for a visual of this path.

We can use this equation:

[tex]\displaystyle v=\frac{d}{t}[/tex]

We are given the velocity and time of the airplane; therefore, we can solve for d in the equation.

[tex]\displaystyle d=vt[/tex]
[tex]\displaystyle d=(115)(15)[/tex]
[tex]d=1725[/tex]

The distance that the airplane travels is 1725 m.

Now, using our knowledge we know that the circumference of a circle is 2πr, and this is the distance traveled for an object in a circular motion.

However, in this problem, the airplane only travels for a fourth of a circular path, so we can use [tex]\displaystyle \frac{2\pi r}{4}[/tex]. This gives us 1/4 the circumference of a circle.

This is equal to the distance (1725 m) the airplane travels:

[tex]\displaystyle 1725=\frac{2\pi r}{4}[/tex]

Multiply 4 to both sides of the equation.

[tex]6900=2\pi r[/tex]

Divide both sides by 2pi.

[tex]r=1098.169107[/tex]
[tex]r \approx 1100[/tex]

The radius of the curve that the plane follows in making the turn is about 1100 m.