a bed is in a form of a circular sector. find its radius if the area of the sector is 1000 centimeter square and its perimeter is a minimum
a = 1/2 r^2 θ
so,
θ = 2000/r^2
p = 2r + rθ = 2r + 2000/r
so, find where dp/dr=0
To find the radius of the circular sector, we need to optimize the perimeter while keeping the area of the sector fixed at 1000 square centimeters.
Let's break down the problem into steps:
Step 1: Write down the formulas for the area and perimeter of a circular sector.
- Area of a circular sector: A = (θ/360) * π * r^2
- Perimeter of a circular sector: P = 2πr + 2s
Note: In the perimeter formula, s represents the length of the arc, and θ is the corresponding angle in degrees.
Step 2: Substitute the given values into the area equation.
1000 = (θ/360) * π * r^2
Step 3: Simplify the equation.
Divide both sides by (θ/360) * π:
1000 / [(θ/360) * π] = r^2
Step 4: Solve for r.
Take the square root of both sides to get the value of r:
r = √[(1000 / [(θ/360) * π])]
Step 5: Optimize the perimeter.
To minimize the perimeter, we need to find the value of θ that minimizes P. The value of θ will be between 0° and 360°.
Step 6: Substitute the value of r into the perimeter equation.
P = 2πr + 2s
P = 2π√[(1000 / [(θ/360) * π])] + 2s
Step 7: Differentiate the perimeter equation with respect to θ.
To find the minimum of P, we differentiate it with respect to θ and set it equal to zero.
dP/dθ = 0
Step 8: Solve the differential equation.
Solving the equation dP/dθ = 0 will give us the value of θ that minimizes the perimeter. However, the process of solving and differentiating the equation might require additional information about the relationship between s and the angle θ.
Once you have this additional information, you can solve the differential equation, find the value of θ, and substitute it into the expression for r to get the optimized radius.