A sector AOB of a circle, centre O, radius r cm, where the acute angle AOB is theta radians. Given that the perimeter of the sector is 14 cm and that the area of the sector is 10 cm^2, evaluate r and theta.
To solve this problem, we need to use some formulas related to the area and perimeter of a sector of a circle.
The perimeter of a sector can be calculated using the formula:
Perimeter of the sector = Length of the arc + 2 * Radius
In this case, the perimeter of the sector is given as 14 cm. So, we have:
14 = Length of the arc + 2 * r
The area of a sector can be calculated using the formula:
Area of the sector = (θ/2) * r^2
In this case, the area of the sector is given as 10 cm^2. So, we have:
10 = (θ/2) * r^2
We now have two equations with two unknowns (r and θ). We can solve these equations simultaneously to find their values.
First, let's rearrange the equation for the perimeter to isolate the length of the arc:
Length of the arc = Perimeter of the sector - 2 * r
Substituting this expression for the length of the arc into the equation for the area, we get:
10 = (θ/2) * r^2
Now, substitute the expression for the length of the arc into the equation for the perimeter:
14 = (Perimeter of the sector - 2 * r) + 2 * r
Simplifying this equation, we get:
14 = Perimeter of the sector
So, the perimeter of the sector is equal to 14.
Now, we have the following two equations:
Perimeter of the sector = 14
Area of the sector = 10
Let's solve these equations to find the values of r and θ.