Part of the stage decoration is a "fan" shape and is to be covered with a white cloth. If the radius of the fan is x+3 and its area is 198 meter squared, what is the dimension of the fan?
The fan is a sector of a circle. That means that
198 = 1/2 (x+3)^2 θ
Unless you know how much of a circle the fan occupies (that is, what is θ, compared to 2π), there's no way to solve for x.
If the fan is a half-circle, θ=π, so
198 = 1/2 (x+3)^2 (π/2)
(x+3)^2 = 198*4/π = 252.10
x+3 = 15.88
x = 12.88
Now use that to decide what the "dimension" is.
oops - I plugged in π/2 instead of π.
The solution above is correct if the fan is 1/4 of a circle.
To find the dimensions of the fan, we need to determine the length of the radius (x+3). We are given that the area of the fan is 198 square meters.
The formula to calculate the area of a circle is A = πr^2, where A represents the area and r represents the radius.
Since we know the area (198) and the radius (x+3), we can set up the equation as follows:
198 = π(x+3)^2
To solve for the dimension of the fan, we need to isolate x. Here's how you can do it:
1. Start by expanding the equation:
198 = π(x^2 + 6x + 9)
2. Simplify further by distributing π:
198 = πx^2 + 6πx + 9π
3. Move all the terms to one side of the equation to set it equal to zero:
πx^2 + 6πx + 9π - 198 = 0
4. Now we have a quadratic equation in terms of x. To solve, you can either factor it or use the quadratic formula. I'll demonstrate using the quadratic formula.
5. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = π, b = 6π, and c = (9π - 198). Substituting these values, we get:
x = (-6π ± √((6π)^2 - 4(π)(9π - 198))) / (2π)
6. Simplify the equation further. First, calculate the values inside the square root:
x = (-6π ± √(36π^2 - 36π^2 + 792π)) / (2π)
x = (-6π ± √(792π)) / (2π)
7. Cancel out the common factors, π:
x = (-6 ± √792) / 2
8. Now we can simplify further. √792 is approximately 28.1425, so we have:
x = (-6 ± 28.1425) / 2
The two possible solutions for x are:
a) x = (-6 + 28.1425) / 2 ≈ 11.071
b) x = (-6 - 28.1425) / 2 ≈ -17.071
Since we are dealing with the radius, the dimension cannot be negative. Therefore, the dimension of the fan is x + 3 ≈ 11.071 + 3 ≈ 14.071 units.