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=36
The equation of the cylinder x2+y2=36 represents a circle in the xy-plane with radius 6 centered at the origin.
A standard parametrization of this circle is:
x=6cos(t),y=6sin(t),
where t is the parameter, and t∈[0,2π).
Step 2: Use z=xy to find z
The surface z=xy relates z to the x and y coordinates.
Substitute the parametric expressions for x and y into z=xy:
z=(6cos(t))(6sin(t))=36cos(t)sin(t).
Using the trigonometric identity sin(2t)=2sin(t)cos(t), we can rewrite z as:
z=18sin(2t).
Step 3: Write the vector function
Combine the parametric expressions for x, y, and z into a single vector function:
r(t)=⟨x(t),y(t),z(t)⟩=⟨6cos(t),6sin(t),18sin(2t)⟩.
Final Answer:
r(t)=⟨6cos(t),6sin(t),18sin(2t)⟩
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