What is the maximum area of a sector of a circle with P = 100? (Here is another surprise!!! -- could you believe it is r2 when r = 25? How is this similar to a square being the maximum rectangle and the central angle of the maximum sector being 2 radians?)

To find the maximum area of a sector of a circle, we need to consider the relationship between the central angle (θ) and the radius (r). The formula to calculate the area of a sector is given by:

Area = (θ/360) * π * r^2

Given that the perimeter (P) of the sector is 100, we can use it to find the value of the central angle (θ). The perimeter of a sector is the sum of the arc length and the two radii, so we have:

P = θ * r + 2r

Since we have the value of P (100) and want to find the maximum area, we can use calculus to solve for θ.

Now, for a sector to have the maximum area, the central angle (θ) needs to be as large as possible. However, θ cannot exceed 2π radians (which is equivalent to a full circle or 360 degrees). This is because a sector cannot have an angle greater than the entire circle.

So, we want to find the maximum area while keeping the central angle (θ) under 2π radians. Let's proceed to calculate the maximum area for a given value of r.

Given r = 25, we can substitute this value in the equation for the sector's perimeter:

100 = θ * 25 + 2 * 25
100 = θ * 25 + 50
50 = θ * 25
θ = 50/25 = 2 radians

Since θ is equal to 2 radians, we can proceed to calculate the maximum area using the formula mentioned earlier.

Area = (θ/360) * π * r^2
Area = (2/360) * π * 25^2
Area = (1/180) * π * 625
Area ≈ 3.47 square units (rounded to two decimal places)

Therefore, the maximum area of the sector is approximately 3.47 square units when the radius (r) is 25 and the central angle (θ) is 2 radians.