We can use the area of the fan and the angle formed to find the length of the fan's edge.
The area of a triangle is given by the formula:
area = (1/2) * base * height
In this case, the base of the triangle is the length of the fan's edge, and the height is the distance from the fixed end to the free end of the fan. Let's call this distance h.
We can find h using trigonometry. The height h is the adjacent side of the angle formed, and the length of the fan's edge is the hypotenuse. Therefore, we can use the cosine function:
cos(50°) = adjacent / hypotenuse
cos(50°) = h / L (where L is the length of the fan's edge)
Solving for h:
h = L * cos(50°)
Now we can substitute this value for h in the formula for the area:
20 cm² = (1/2) * L * L * cos(50°)
Simplifying:
40 cm² = L² * cos(50°)
Dividing both sides by cos(50°):
L² = 40 cm² / cos(50°)
Taking the square root of both sides:
L = sqrt(40 cm² / cos(50°))
Using a calculator, we get:
L ≈ 8.88 cm
Therefore, the length of the fan's edge is approximately 8.88 cm.