An ant starts at the point (3,0) on a circle of radius 3 meters and walks counterclockwise a distance of 6 meters around the circle. Find the x and the y coordinates of the ants location.
Round to the nearest hundredth.
draw a line from his final position to the centre and complete the triangle
all the way around is 2πr = 6π
so completed 6/6π or 1/π around the circle
angle"
Ø/360 = 6/6π
Ø = 360/π
= 114.59156°
cosØ = x/3 and sinØ = y/3
x = 3cosØ = -1.25
y = 3sinØ = 2.73
check
(-1.25)^2 + 2.73^2 = 9
radius = √9 = 3
To find the x and y coordinates of the ant's location after walking counterclockwise a distance of 6 meters around the circle, we can use the concept of radians.
First, let's find the angle in radians that corresponds to a distance of 6 meters around the circle. The circumference of a circle is given by the formula 2πr, where r is the radius. In this case, the radius is 3 meters. So the circumference is 2π * 3 = 6π meters. Since the ant walks a distance of 6 meters, the angle it covers is 6/6π = 1/π radians.
Now, let's use trigonometry to find the x and y coordinates. Recall that the x-coordinate on a circle is given by r * cos(θ) and the y-coordinate is given by r * sin(θ), where θ is the angle in radians.
In this case, the radius is 3 meters and the angle is 1/π radians. Therefore, the x-coordinate is 3 * cos(1/π) and the y-coordinate is 3 * sin(1/π).
Calculating these values:
x-coordinate = 3 * cos(1/π) ≈ 3 * 0.318 ≈ 0.954
y-coordinate = 3 * sin(1/π) ≈ 3 * 0.949 ≈ 2.847
Therefore, the approximate x-coordinate of the ant's location is 0.954 and the approximate y-coordinate is 2.847.
To find the coordinates of the ant's location, we'll use trigonometry and the concept of radians.
First, let's visualize the situation. The ant starts at the point (3, 0) on a circle of radius 3 meters. It then walks counterclockwise for a distance of 6 meters around the circle.
Since the ant walks a distance of 6 meters, we can calculate the angle it covers using the formula:
angle = distance / radius
Substituting the values, we have:
angle = 6 / 3
angle = 2 radians
Now, let's convert this angle to find the new coordinates of the ant.
In trigonometry, the coordinates on a circle with radius r can be expressed as:
(x, y) = (r * cos(angle), r * sin(angle))
Substituting the values, we have:
(x, y) = (3 * cos(2), 3 * sin(2))
Using a calculator, we find:
cos(2) ≈ -0.416
sin(2) ≈ 0.909
Substituting these values, we get:
(x, y) ≈ (3 * -0.416, 3 * 0.909)
(x, y) ≈ (-1.249, 2.727)
Therefore, the approximate x-coordinate of the ant's location is -1.25, and the approximate y-coordinate is 2.73.