Γ is a circle with center O. A and B are points on Γ such that the sector AOB has a perimeter of 40. Amongst all circular sectors with a perimeter of 40, what is the central measure of ∠AOB (in radians) of the sector with the largest area?

To find the central measure of the angle ∠AOB in radians for the sector with the largest area, we need to determine the radius of the circle Γ first.

Let's assume the radius of the circle Γ is "r". Since A and B are points on the circumference of the circle, the length of the arc AB is equal to the perimeter of the sector AOB.

Given that the sector AOB has a perimeter of 40, it means AB = 40.

The formula to calculate the length of an arc is: arc length = (central angle / 2π) * (2π * r).

In this case, the length of the arc AB is equal to (central angle / 2π) * (2π * r), which can be written as:

40 = (central angle / 2π) * (2π * r)

Simplifying this equation, we have:

40 = central angle * r

Now, to find the central angle that maximizes the area of the sector, we need to maximize the area of the sector. The formula for the area of a sector is:

Area = (1/2) * (r^2) * central angle

Since we want to maximize the area, we want to maximize the central angle while keeping the perimeter constant.

From the equation 40 = central angle * r, we can see that as r increases, the central angle must decrease to keep the product constant.

To maximize the area of the sector, we want the central angle to be as large as possible, which occurs when r is minimized.

In other words, the smallest possible radius will result in the largest central angle for a given perimeter.

To minimize the radius, we can consider the extreme case where the arc AB forms a semicircle. In this case, the central angle is π radians.

Substituting π for the central angle in the equation 40 = central angle * r gives:

40 = π * r

Solving for r, we have:

r = 40 / π

Now that we know the radius "r", we can find the central measure of ∠AOB in radians. Since the arc AB is the entire circumference of the circle, the central angle of ∠AOB is 2π radians.

Therefore, the central measure of ∠AOB for the sector with the largest area is 2π radians.