I wont be receiving credit for these problems but i was just wondering how would i answer them...
A) The area of a sector of a circle with a central angle of 25π rad is 21 mm².
B) The angle between 0 and 2π in radians that is coterminal with the angle 487π in radians is...
C) Find the distance that the earth travels in one day in its path around the sun. Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius 93 million miles.
For B... i meant to say 48/7 pi not 487pi
A)
a = 1/2 r^2θ, so
21 = 1/2 r^2 25π
I gotta wonder about an angle of 25π, since 2π is a complete circle...
B)
487π = 243(2π) + π
C)
1/365 * 2πr
A) To find the area of a sector of a circle, you can use the formula:
Area = (θ/360) * π * r^2
where θ is the central angle in degrees, π is a constant approximately equal to 3.14159, and r is the radius of the circle.
In this case, the central angle is given as 25π radians, and the area is given as 21 mm².
Since the formula requires the central angle to be in degrees, we need to convert the central angle from radians to degrees. We know that 2π radians is equivalent to 360 degrees. Therefore, we can set up a proportion to find the central angle in degrees:
25π radians = x degrees
2π radians = 360 degrees
Cross-multiplying, we get:
25π * 360 = 2π * x
Simplifying, we find:
9000π = 2π * x
Canceling out the π:
9000 = 2x
x = 4500
So the central angle in degrees is 4500 degrees.
Now, we can plug the values into the formula to find r (radius). The area is given as 21 mm².
21 = (4500/360) * π * r^2
Simplifying, we have:
21 = (12.5) * π * r^2
Dividing both sides by 12.5π:
r^2 = 21 / (12.5π)
Taking the square root of both sides, we find:
r ≈ √(21 / (12.5π))
Therefore, the radius of the circle is approximately equal to √(21 / (12.5π)).
B) To find the coterminal angle, we need to find another angle with the same initial and terminal sides. Coterminal angles can be found by adding or subtracting a multiple of 2π.
In this case, the given angle is 487π radians. To find an equivalent angle between 0 and 2π, we can subtract or add a multiple of 2π.
To subtract a multiple of 2π, we divide the given angle by 2π and then multiply the result by 2π:
487π - (2 * π) = 483π
Therefore, the angle between 0 and 2π in radians that is coterminal with the angle 487π in radians is 483π.
C) To find the distance that the earth travels in one day in its path around the sun, we need to calculate the circumference of the circular path. The circumference of a circle is given by the formula:
Circumference = 2 * π * r
where π is a constant approximately equal to 3.14159, and r is the radius of the circle.
In this case, the radius of the circular path is given as 93 million miles.
So, the distance that the earth travels in one day is:
Distance = Circumference * Number of Days
Given that a year has 365 days, we can substitute the values into the formula:
Distance = (2 * π * 93 million) * 365
Simplifying, we have:
Distance = 2π * 93 million * 365
Calculating this value will give you the distance that the earth travels in one day in its path around the sun.