Please Help D:. Min/Max Problems

4. Joan has 50ft of interlocking stone available of fencing off a flower garden in the form of a circular sector. Find the radius of the circle that will yield a flower garden with the largest are if Joan uses all the stone.

The perimeter of the garden is

p = 2r + rθ, so
50 = r(2+θ), and θ = 50/r - 2

the area is
a = 1/2 r^2 θ
= 1/2 r^2 (50/r - 2)
= 25r - r^2
= r(25-r)

This is just a parabola with vertex at r=25/2, so that's the radius giving maximum area

To solve this problem, we can use the concepts of calculus and optimization. Let's break it down into steps:

Step 1: Understand the problem
Joan wants to fence off a flower garden in the form of a circular sector using 50ft of interlocking stone. She wants to find the radius of the circle that will yield the largest area if she uses all the stone.

Step 2: Identify the variables
Let's assign some variables:
- r: the radius of the circle
- A: the area of the circular sector
- C: the circumference of the circular sector

Step 3: Formulate the problem
The area of the circular sector can be calculated using the formula A = (π * r^2 * θ) / 360, where θ is the central angle.

The circumference of the circular sector can be calculated using the formula C = 2πr, and we know that C = 50ft.

Step 4: Express the area in terms of a single variable
Since we need to find the maximum area, we want to express the area A in terms of a single variable. We can do this by eliminating θ from the equation.

Using the relationship between C and θ, we can express θ as (360 * C) / (2πr).

Substitute the value of θ in the area formula:
A = (π * r^2 * ((360 * C) / (2πr))) / 360
Simplify: A = (1800C) / r

Step 5: Find the derivative of A with respect to r
To find the maximum area, we need to find the value of r that maximizes A. To do this, we will find the derivative of A with respect to r and set it equal to zero.

dA/dr = -1800C / r^2 = 0

Step 6: Solve for r
Solving the equation, we get:
r = sqrt(1800C)

Step 7: Substitute the value of C
We know that C = 50ft, so substitute it into the equation:
r = sqrt(1800 * 50) = sqrt(90000) = 300ft

Therefore, the radius of the circle that will yield the largest area is 300ft.