A sector in a circle of radius 8.1 m has an area of 53.5 m^2. The central angle in a sector must be _____o. Round to one decimal place.
Hint:
a sector is a pie-piece, similar to a slice of pizza.
The area of a sector of a given radius is proportional to the central angle.
Area of Sector
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Area of circle
central angle
=-------------
360°
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To find the central angle in a sector, we need to use the formula:
Area of sector = (θ/360) * π * r^2,
where θ is the central angle, π (pi) is a constant approximately equal to 3.14159, and r is the radius of the circle.
In this case, we are given the area of the sector, which is 53.5 m^2, and the radius, which is 8.1 m. We can substitute these values into the formula and solve for θ:
53.5 = (θ/360) * π * (8.1^2),
To isolate θ, we can first rearrange the equation:
θ/360 = 53.5 / (π * (8.1^2)).
Next, we can multiply both sides by 360 to get rid of the fraction:
θ = (53.5 / (π * (8.1^2))) * 360.
Now we can use a calculator to simplify and calculate the value of θ:
θ ≈ 85.0 degrees (rounded to one decimal place).
Therefore, the central angle in the sector is approximately 85.0 degrees.