Find a parametrization, using cos(t)
and sin(t)
, of the following curve:
The intersection of the plane y=6
with the sphere x2+y2+z2=117
r(t)=
To find a parametrization of the curve, we need to find the values of x, y, and z that satisfy both the equation of the plane and the equation of the sphere.
First, let's consider the equation of the plane: y = 6.
Since y is a constant 6, we can substitute this value into the equation of the sphere to find the values of x and z that satisfy both equations.
x^2 + y^2 + z^2 = 117
x^2 + 6^2 + z^2 = 117
x^2 + 36 + z^2 = 117
x^2 + z^2 = 81
Now, let's rewrite this equation in terms of x and z:
x^2 = 81 - z^2
To create a parametrization using cos(t) and sin(t), we need to express x and z in terms of these trigonometric functions.
Since the equation x^2 = 81 - z^2 involves both x and z, we need to find a relationship between x and z that involves sin(t) and cos(t).
To do this, we can choose a convenient angle t such that sin(t) and cos(t) can be used to represent x and z.
Let's choose t such that x = 9cos(t) and z = 9sin(t).
Substituting these values into the equation x^2 + z^2 = 81, we get:
(9cos(t))^2 + (9sin(t))^2 = 81
81cos^2(t) + 81sin^2(t) = 81
81(cos^2(t) + sin^2(t)) = 81
81(1) = 81
This equation is true, so x = 9cos(t) and z = 9sin(t) satisfy the equation x^2 + z^2 = 81.
Now, we can substitute these values into the equation of the plane y = 6 to find the value of y that satisfies the equation.
Since y = 6, the parametrization of the curve is:
r(t) = (x, y, z)
= (9cos(t), 6, 9sin(t))
To find a parametrization of the curve that represents the intersection of the plane y=6 and the sphere x^2 + y^2 + z^2 = 117, we can start by substituting the value of y into the equation of the sphere:
x^2 + (6)^2 + z^2 = 117
Simplifying this equation:
x^2 + 36 + z^2 = 117
x^2 + z^2 = 81
This equation represents a circle in the x-z plane centered at the origin with a radius of sqrt(81) = 9. We can parametrize this circle using cosine and sine functions:
x(t) = 9 * cos(t)
z(t) = 9 * sin(t)
Since y is fixed at 6, the parametrization of the curve is:
r(t) = (x(t), 6, z(t)) = (9 * cos(t), 6, 9 * sin(t))
So, the parametrization of the curve is given by r(t) = (9 * cos(t), 6, 9 * sin(t)).