the area of a sector of a circle of radius 8cm is 45cm2. find the size of the angle subtended at the center of the circle, correct to one decimal place. [Take =22/7].
Area of whole circle = π(8^2) = 64π
For our sector:
64π/45 = 2π radians/ θ radians
θ = 2π/(64π)(45) = 45/32 radians, which is 80.57°
The bot had
45 = (θ/360)(22/7)(8^2)
which would give the answer in degrees, but it stated it as radians.
It also made an error in the arithmetic to get θ = 2.29
Thank you for the correction! You are absolutely right. I apologize for my mistake in my previous response. Thank you for catching that and providing the correct solution.
AAAaannndd the bot gets it wrong yet again!
A = 1/2 r^2 θ so
1/2 * 8^2 θ = 45
32θ = 45
θ = 45/32 = 1.40 radians
I apologize again for the mistake in my previous response. You are absolutely right in the method you used to solve the problem. Thank you for providing the correct solution.
To find the size of the angle subtended at the center of the circle, we can use the formula for the area of a sector of a circle:
Area of sector = (θ/360) * π * r^2
Given:
Area of sector = 45 cm^2
Radius (r) = 8 cm
π (pi) = 22/7
Let's substitute the given values into the formula and solve for θ (the angle subtended at the center):
45 = (θ/360) * (22/7) * (8^2)
First, simplify the equation:
45 = (θ/360) * (22/7) * 64
45 = (θ/360) * 22 * 64/7
45 = (θ/360) * 1408/7
45 = (θ/360) * 201.14
To isolate θ, let's cross multiply and solve for θ:
45 * 360 = θ * 201.14
16,200 = θ * 201.14
Now, divide both sides by 201.14 to solve for θ:
θ = 16,200 / 201.14
θ ≈ 80.6 (rounded to one decimal place)
Therefore, the size of the angle subtended at the center of the circle is approximately 80.6 degrees.
We know that the formula for the area of a sector is A = (θ/360)πr^2, where θ is the angle subtended at the center of the circle in degrees.
Let's plug in the values we know: A = 45, r = 8, and π = 22/7.
45 = (θ/360)(22/7)(8^2)
45 = (θ/360)(22/7)(64)
45 = (θ/5)(88)
θ/5 = 45/88
θ = (45/88)*5
θ ≈ 2.29 radians
Now we need to convert this angle to degrees by multiplying by 180/π:
θ ≈ (2.29)*(180/π)
θ ≈ 131.3 degrees
Therefore, the size of the angle subtended at the center of the circle is approximately 131.3 degrees.