Matilda is planning a walk around the perimeter of Wedge Park, which is shaped like a circular wedge. The walk around the park is 2.1 miles, and the park has an area of 0.25 square miles. If θ is less than 90 degrees, what is the value of the radius.
area wedge= 2PI r^2*theta/360 where theta is in degrees
perimeter= 2r+ theta/360 * 2PI r
Can you take it from here?
To find the value of the radius of Wedge Park, we can use the formula for the circumference of a circle:
C = 2πr
where C is the circumference and r is the radius.
Given that the walk around the park is 2.1 miles, we can set up the equation:
2.1 miles = 2πr
Next, let's find the value of the angle θ in degrees. Since Wedge Park is shaped like a circular wedge, the angle formed by the circular arc is related to the area of the park. The area of a circular sector is given by the formula:
A = (θ/360) × πr^2
Given that the area of the park is 0.25 square miles, we can set up the equation:
0.25 square miles = (θ/360) × πr^2
Now we have two equations:
2.1 miles = 2πr --> Equation 1
0.25 square miles = (θ/360) × πr^2 --> Equation 2
Since θ is less than 90 degrees, θ/360 can be expressed as a decimal less than 1.
To solve for the radius, we need to use both equations simultaneously. We can rearrange Equation 1 to solve for r:
r = 2.1/(2π)
Plugging this value of r into Equation 2, we can solve for θ:
0.25 = (θ/360) × π(2.1/(2π))^2
Simplifying:
0.25 = (θ/360) × ((2.1/2)^2)
0.25 = (θ/360) × 1.05^2
0.25 = (θ/360) × 1.1025
Now we can solve for θ by multiplying both sides of the equation by 360 and dividing by 1.1025:
θ = (0.25 * 360) / 1.1025
θ ≈ 82.208
So the value of the angle θ is approximately 82.208 degrees.
Finally, we can substitute this value of θ back into Equation 1 to find the value of the radius:
2.1 miles = 2πr
2.1 miles = 2πr
r = 2.1 miles / (2π)
r ≈ 0.333 miles
Therefore, the value of the radius is approximately 0.333 miles.