At what points does the helix r = sin(t), cos(t), t intersect the sphere x 2 + y 2 + z 2 = 37? (Round your answers to three decimal places. Enter your answers from smallest to largest z-value.)
Well, let's see if we can find where the helix intersects with the sphere. Remember that the equation of the helix is r = sin(t), cos(t), t, and the equation of the sphere is x^2 + y^2 + z^2 = 37.
First, we can substitute the equation of the helix into the equation of the sphere to get:
(sin(t))^2 + (cos(t))^2 + t^2 = 37.
Since (sin(t))^2 + (cos(t))^2 = 1, we can simplify the equation to:
1 + t^2 = 37.
Subtracting 1 from both sides gives us:
t^2 = 36.
Taking the square root of both sides, we find that t = ± 6.
So the helix intersects the sphere at t = -6 and t = 6.
Now, let's substitute these values of t back into the equation of the helix to find the corresponding points:
When t = -6:
r = sin(-6), cos(-6), -6 = -0.279, 0.961, -6.
When t = 6:
r = sin(6), cos(6), 6 = -0.279, 0.961, 6.
So, the helix intersects the sphere at (-0.279, 0.961, -6) and (-0.279, 0.961, 6).
Remember to round your answers to three decimal places and list them from smallest to largest z-value.
So, the intersection points are (-0.279, 0.961, -6) and (-0.279, 0.961, 6).
To find the points of intersection between the helix and the sphere, we need to solve the following system of equations:
r = sin(t),
r = cos(t),
x^2 + y^2 + z^2 = 37.
Let's proceed step by step:
Step 1: Equate the x-coordinate of the helix (r = sin(t)) with x in the sphere equation:
sin(t) = x. (Equation 1)
Step 2: Equate the y-coordinate of the helix (r = cos(t)) with y in the sphere equation:
cos(t) = y. (Equation 2)
Step 3: Substitute the x and y values from Equations 1 and 2 into the sphere equation:
x^2 + y^2 + z^2 = 37.
sin(t)^2 + cos(t)^2 + z^2 = 37.
1 + z^2 = 37.
z^2 = 36.
z = ±6.
So, the z-values of the points of intersection are -6 and 6.
Hence, the points are (sin(t), cos(t), 6) and (sin(t), cos(t), -6), where t is any real number.
These are the points of intersection between the helix and the sphere.
To find the points where the helix intersects the sphere, we need to solve the system of equations formed by the parametric equations of the helix and the equation of the sphere.
The parametric equations of the helix are:
x = sin(t)
y = cos(t)
z = t
The equation of the sphere is:
x^2 + y^2 + z^2 = 37
Let's substitute the parametric equations of the helix into the equation of the sphere to get a single equation in terms of t:
(sin(t))^2 + (cos(t))^2 + t^2 = 37
Simplifying this equation, we have:
sin^2(t) + cos^2(t) + t^2 = 37
1 + t^2 = 37
t^2 = 36
t = ± 6
Now, substitute these values of t back into the parametric equations of the helix to find the corresponding points of intersection:
For t = 6:
x = sin(6) ≈ 0.566
y = cos(6) ≈ 0.828
z = 6
For t = -6:
x = sin(-6) ≈ -0.566
y = cos(-6) ≈ 0.828
z = -6
Therefore, the two points where the helix intersects the sphere are approximately:
(0.566, 0.828, 6) and (-0.566, 0.828, -6).
A helix r=(sin(t),cos(t),t) is a coil like a spring of radius 1 around the z-axis.
The coordinates in cartesian coordinates are:
x(t)=sin(t)
y(t)=cos(t)
z(t)=t
The distance from origin is
r=√(sin²(t)+cos²(t)+t²)
When it intersects a sphere of radius R, the following relationship applies:
sin²(t)+cos²(t)+t² = 37² ...(1)
Using sin²(x)+cos²(x)=1, (1) simplifies to
1+t²=37² ...(2)
Solve 2 for t and back substitute to find x(t), y(t) and z(t)