Write the equations in cylindrical coordinates.
(a)
3z = 2x^2 + 2y^2
(b)
7x^2 + 7y^2 = 4y
(a) Sure, here you go:
In cylindrical coordinates, the equation 3z = 2x^2 + 2y^2 becomes:
3z = 2ρ^2 (since x = ρcosθ and y = ρsinθ)
(b) Now let's tackle the equation 7x^2 + 7y^2 = 4y:
In cylindrical coordinates, the equation 7x^2 + 7y^2 = 4y becomes:
7(ρcosθ)^2 + 7(ρsinθ)^2 = 4ρsinθ
Simplifying it further, we have:
7ρ^2(cos^2θ + sin^2θ) = 4ρsinθ
7ρ^2 = 4ρsinθ
Hope that brings a smile to your face!
To write the equations in cylindrical coordinates, we need to express the variables x, y, and z in terms of the cylindrical coordinates ρ, φ, and z.
(a) 3z = 2x^2 + 2y^2
To express x and y in terms of cylindrical coordinates, we use the following conversions:
x = ρ*cos(φ)
y = ρ*sin(φ)
z = z
Substituting these equations into the given equation:
3z = 2(ρ*cos(φ))^2 + 2(ρ*sin(φ))^2
3z = 2ρ^2*cos^2(φ) + 2ρ^2*sin^2(φ)
Simplifying further:
3z = 2ρ^2*(cos^2(φ) + sin^2(φ))
3z = 2ρ^2
The equation in cylindrical coordinates is 3z = 2ρ^2.
(b) 7x^2 + 7y^2 = 4y
Using the cylindrical coordinate conversions, we have:
x = ρ*cos(φ)
y = ρ*sin(φ)
z = z
Substituting these values into the given equation:
7(ρ*cos(φ))^2 + 7(ρ*sin(φ))^2 = 4(ρ*sin(φ))
Simplifying further:
7ρ^2*cos^2(φ) + 7ρ^2*sin^2(φ) = 4ρ*sin(φ)
7ρ^2*(cos^2(φ) + sin^2(φ)) = 4ρ*sin(φ)
Since cos^2(φ) + sin^2(φ) = 1, we can simplify further:
7ρ^2 = 4ρ*sin(φ)
Dividing both sides by ρ:
7ρ = 4*sin(φ)
The equation in cylindrical coordinates is 7ρ = 4*sin(φ).
To express the equations in cylindrical coordinates, we need to replace the Cartesian variables (x, y, z) with their equivalent cylindrical variables (ρ, φ, z).
(a) 3z = 2x^2 + 2y^2
In cylindrical coordinates, x = ρcos(φ), y = ρsin(φ), and z = z. Substituting these values into the given equation, we get:
3z = 2(ρcos(φ))^2 + 2(ρsin(φ))^2
Simplifying this equation further:
3z = 2ρ^2cos^2(φ) + 2ρ^2sin^2(φ)
Using the trigonometric identity cos^2(φ) + sin^2(φ) = 1:
3z = 2ρ^2(1)
Therefore, the equation in cylindrical coordinates is: 3z = 2ρ^2.
(b) 7x^2 + 7y^2 = 4y
Similarly, substituting x = ρcos(φ) and y = ρsin(φ) into the equation, we have:
7(ρcos(φ))^2 + 7(ρsin(φ))^2 = 4(ρsin(φ))
Simplifying:
7ρ^2cos^2(φ) + 7ρ^2sin^2(φ) = 4ρsin(φ)
Using the trigonometric identity cos^2(φ) + sin^2(φ) = 1:
7ρ^2(1) = 4ρsin(φ)
Therefore, the equation in cylindrical coordinates is: 7ρ^2 = 4ρsin(φ).
(a) z = 2/3 r^2
(b)
7r^2 = 4r sinθ
r = 4/7 sinθ