A Bed is in the form of a circular sector of radius and central angle θ. Find its radius if the area of the sector is 100 cm2 and its perimeter is a minimum.
A = 1/2 r^2 θ = 100
The perimeter p is
p = 2r + rθ = r(θ+2) = r(200/r^2 + 2) = 200/r + 2r
dp/dr = 2 - 200/r^2 = 2/r^2 (r^2-100)
so p is a minimum when r = 10
To find the radius of the circular sector, we need to minimize its perimeter.
Let's define the radius of the circular sector as r and the central angle as θ.
The area of a circular sector can be calculated using the formula:
Area = (θ/360) * π * r^2
From the given information, we have:
Area = 100 cm^2
So, we can write the equation as:
(θ/360) * π * r^2 = 100
Now, the perimeter of the circular sector consists of the curved part (arc) and the two radii.
Perimeter = Arc Length + 2 * r
The formula to calculate the arc length is given by:
Arc Length = (θ/360) * 2 * π * r
So, the perimeter equation becomes:
Perimeter = (θ/360) * 2 * π * r + 2 * r
To find the minimum perimeter, we need to differentiate this equation with respect to r and set it equal to zero:
d(Perimeter)/dr = (θ/360) * 2 * π + 2 = 0
Simplifying this equation, we have:
(θ/360) * 2 * π + 2 = 0
Dividing both sides by 2 and rearranging, we get:
(θ/360) * π = -1
Now, substitute the value of θ/360 from this equation into the area equation we had earlier:
(-1) * π * r^2 = 100
Simplifying, we get:
r^2 = -100/π
Taking the square root of both sides, we have:
r = √(-100/π)
Since the square root of a negative value is not possible in this context, the value of r does not exist for the given conditions.
Therefore, it is not possible to determine the radius of the circular sector that satisfies the given area and perimeter requirements.
To find the radius of the circular sector, we need to minimize its perimeter.
The perimeter of a circular sector consists of two radii (the straight sides of the sector) and an arc (the curved part of the sector).
The length of the arc can be calculated using the formula:
arc length = (θ / 360) * 2πr, where θ is the central angle and r is the radius of the sector.
Let's calculate the arc length using the given area of the sector. The area of a circular sector is given by:
sector area = (θ / 360) * π * r^2, where θ is the central angle and r is the radius of the sector.
We have been given that the sector area is 100 cm^2. Substituting this value in the formula above, we get:
100 = (θ / 360) * π * r^2.
Now, let's express θ in terms of r and solve for θ:
θ = (100 * 360) / (π * r^2).
Substituting this value of θ in the arc length formula, we get:
arc length = ((100 * 360) / (π * r^2) / 360) * 2πr
= (100 / π * r^2) * 2πr
= (200r / π).
The perimeter of the circular sector is given by:
perimeter = 2r + arc length
= 2r + (200r / π).
To minimize the perimeter, we can take the derivative of the perimeter with respect to r, set it equal to zero, and solve for r.
d(perimeter) / dr = 2 + (200 / π) = 0
Simplifying the equation, we get:
2r + (200r / π) = 0
Multiplying through by π, we obtain:
2πr + 200r = 0
Combining like terms:
(2π + 200)r = 0
Dividing both sides by (2π + 200), we find:
r = 0 / (2π + 200)
r = 0
Since a radius of zero doesn't make sense in this context, we can conclude that the minimum perimeter of the circular sector does not exist.
Therefore, there is no radius that will minimize the perimeter of the circular sector with an area of 100 cm^2.