The parametric equations for the arc of a circle with a radius of 7, going from (0,0) to (14,0), are:
x = 7 × cos(θ)
y = 7 × sin(θ)
where θ varies from 0 to π (0 to 180 degrees). The circle doesn't reach the point (14,0) along the x-axis, as it only spans half of its circumference.
To find parametric equations for the arc of a circle with a radius of 7 that goes from the point (0,0) to (14,0), you can use the equation of a circle centered at the origin:
x² + y² = r²
Where:
- (x, y) are the coordinates on the circle.
- r is the radius of the circle.
In this case, r = 7. So, the equation becomes:
x² + y² = 7²
x² + y² = 49
Now, we need to parameterize this equation. You can use the trigonometric functions sine and cosine to represent the points on the circle. The parameterization will be:
x = r × cos(θ)
y = r × sin(θ)
Since the circle goes from (0,0) to (14,0), θ will vary from 0 to π radians (or 0 to 180 degrees), because we are moving from the positive x-axis to the negative x-axis.
Now, let's find the parametric equations:
1. For x:
x = 7 × cos(θ)
2. For y:
y = 7 × sin(θ)
Since we want the circle to go from (0,0) to (14,0), we need to find the values of θ that correspond to these two points.
When (x, y) = (0,0), we have:
0 = 7 × cos(θ)
cos(θ) = 0
This happens when θ = π/2 (90 degrees) and θ = 3π/2 (270 degrees).
When (x, y) = (14,0), we have:
14 = 7 × cos(θ)
cos(θ) = 2
However, the cosine function only takes values between -1 and 1, so there are no real values of θ that satisfy this equation. This means that the circle doesn't actually reach the point (14,0) along the x-axis.
So, the parametric equations for the arc of the circle from (0,0) to (14,0) are:
x = 7 × cos(θ)
y = 7 × sin(θ)
with θ ranging from 0 to π (0 to 180 degrees).