At what points does the helix r = sin(t), cos(t), t intersect the sphere x^2 + y^ 2 + z^2 = 26?
(Round your answers to three decimal places. Enter your answers from smallest to largest z-value.)
The sphere, S, is given by:
S(x,y,z): x²+y²+z²-26=0
The helix, r, is given by:
r(x,y,z): sin(t),cos(t),t
which means that the helix is constrained to the cylinder of unit radius, C: C(x,y): x²+y²=1
Thus, the solution for the intersections is given by the solution of
S(sin(t), cos(t), t)=0
from which it can be deduced by inspection that t=5 gives an exact solution, since
(sin²(5)+cos²(5)) + 5²
=(1) + 25
=26
Similarly, t=-5 is a solution.
To find the points of intersection between the helix and the sphere, we need to solve the system of equations formed by equating their respective equations.
Let's set up the equations:
Helix equation:
x = sin(t)
y = cos(t)
z = t
Sphere equation:
x^2 + y^2 + z^2 = 26
Substituting the helix equations into the sphere equation, we have:
(sin(t))^2 + (cos(t))^2 + t^2 = 26
(sin^2(t) + cos^2(t)) + t^2 = 26
1 + t^2 = 26
t^2 = 25
Taking the square root of both sides, we find t = ±5.
Now we can substitute these values of t back into the helix equations to find the corresponding x, y, and z values.
For t = 5:
x = sin(5) ≈ 0.959
y = cos(5) ≈ 0.283
z = 5
For t = -5:
x = sin(-5) ≈ -0.959
y = cos(-5) ≈ 0.283
z = -5
Therefore, the points of intersection between the helix and the sphere are:
Point 1: (x,y,z) ≈ (0.959, 0.283, 5)
Point 2: (x,y,z) ≈ (-0.959, 0.283, -5)
Note: The points are given in order of increasing z-values.
To find the points where the helix intersects the sphere, we need to find the values of t that satisfy both the equations for the helix and the sphere. Let's start by setting up the equations:
The equation of the helix is:
r = sin(t), cos(t), t
The equation of the sphere is:
x^2 + y^2 + z^2 = 26
Now, substitute the values from the helix equation into the equation of the sphere:
(sin(t))^2 + (cos(t))^2 + t^2 = 26
Simplifying this equation will give us the values of t that satisfy both equations. Let's do that:
(sin(t))^2 + (cos(t))^2 + t^2 = 26
(sin(t))^2 + (cos(t))^2 = 26 - t^2
1 = 26 - t^2
t^2 = 25
t = ± 5
So, the values of t that satisfy both equations are t = 5 and t = -5.
Now substitute these values back into the helix equation to find the corresponding points:
For t = 5:
r = sin(5), cos(5), 5
For t = -5:
r = sin(-5), cos(-5), -5
Calculate the values using a calculator or by hand (rounding to three decimal places) to get the points of intersection.
For t = 5:
r = (sin(5), cos(5), 5) ≈ (0.087, 0.996, 5.000)
For t = -5:
r = (sin(-5), cos(-5), -5) ≈ (-0.087, 0.996, -5.000)
So, the two points where the helix intersects the sphere are approximately:
(0.087, 0.996, 5.000) and (-0.087, 0.996, -5.000)