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Find a vector function, r(t)\mathbf{r}(t), that represents the curve of intersection of the two surfaces: The cylinder x2+y2=36x^2 + y^2 = 36 and the surface z=xyz = xy

r(t)=,,\mathbf{r}(t) = \boxed{\langle \dots, \dots, \dots \rangle}

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Question :

Find a vector function, r(t)\mathbf{r}(t), that represents the curve of intersection of the two surfaces: the cylinder x2+y2=36x^2 + y^2 = 36 and the surface z=xyz = xy

r(t)=,,\mathbf{r}(t) = \boxed{\langle \dots, \dots, \dots \rangle}

Find a vector function, \mathbf{r}(t), that represents the curve of intersecti | Doubtlet.com

Solution:

Neetesh Kumar

Neetesh Kumar | December 14, 2024

Calculus Homework Help

This is the solution to Math 1C
Assignment: 13.1 Question Number 10
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Step-by-step solution:

Step 1: Parametrize the cylinder x2+y2=36x^2 + y^2 = 36

The equation of the cylinder x2+y2=36x^2 + y^2 = 36 represents a circle in the xyxy-plane with radius 66 centered at the origin.

A standard parametrization of this circle is:
x=6cos(t),y=6sin(t),x = 6\cos(t), \quad y = 6\sin(t),
where tt is the parameter, and t[0,2π)t \in [0, 2\pi).

Step 2: Use z=xyz = xy to find zz

The surface z=xyz = xy relates zz to the xx and yy coordinates.

Substitute the parametric expressions for xx and yy into z=xyz = xy:
z=(6cos(t))(6sin(t))=36cos(t)sin(t).z = \left(6\cos(t)\right)\left(6\sin(t)\right) = 36\cos(t)\sin(t).
Using the trigonometric identity sin(2t)=2sin(t)cos(t)\sin(2t) = 2\sin(t)\cos(t), we can rewrite zz as:
z=18sin(2t).z = 18\sin(2t).

Step 3: Write the vector function

Combine the parametric expressions for xx, yy, and zz into a single vector function:
r(t)=x(t),y(t),z(t)=6cos(t),6sin(t),18sin(2t).\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle = \langle 6\cos(t), 6\sin(t), 18\sin(2t) \rangle.

Final Answer:

r(t)=6cos(t),6sin(t),18sin(2t)\mathbf{r}(t) = \boxed{\langle 6\cos(t), 6\sin(t), 18\sin(2t) \rangle}


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