Find the maximum value of |r| and any zeros of r. (Assume 0 ≤ θ < 2π. Enter any zeros as a comma-separated list.)
r = 5 + 10cosθ
again,
r' = -10 sinθ
r'=0 at θ = 0,π
you pick which is maximum
To find the maximum value of |r| and any zeros of r, we need to evaluate the expression for different values of θ.
Given: r = 5 + 10cosθ
First, let's find the zeros of r. The zeros occur when cosθ = -0.5. We know that cosθ = -0.5 when θ = 2π/3 and θ = 4π/3.
Therefore, the zeros of r are θ = 2π/3 and θ = 4π/3.
To find the maximum value of |r|, we need to maximize the expression 5 + 10cosθ. Since -1 ≤ cosθ ≤ 1, the maximum value of cosθ is 1.
So, the maximum value of |r| occurs when cosθ = 1. Substituting this into the expression, we have:
|r| = 5 + 10(1)
|r| = 5 + 10
|r| = 15
Therefore, the maximum value of |r| is 15, and the zeros of r are θ = 2π/3 and θ = 4π/3.
To find the maximum value of |r| and any zeros of r, we need to analyze the given equation:
r = 5 + 10cosθ
First, note that the equation is in polar form, where r represents the distance from the origin to a point (x, y) in the Cartesian plane, and θ represents the angle formed with the positive x-axis.
The expression |r| denotes the absolute value of r, which means it's the positive distance from the origin to the point, irrespective of direction.
To find the maximum value of |r|, we need to find the maximum value of r, which occurs when cosθ is minimum (-1).
Substituting cosθ = -1 into the equation, we get:
r = 5 + 10(-1)
r = 5 - 10
r = -5
Thus, the maximum value of |r| is 5.
To find the zeros of r, we set r = 0 and solve for θ. Substitute r = 0 into the equation:
0 = 5 + 10cosθ
Now, solve for cosθ:
10cosθ = -5
cosθ = -1/2
To determine the values of θ for which cosθ is equal to -1/2, we can refer to the unit circle or use inverse trigonometric functions.
We know that cosine is negative in the second and third quadrants of the unit circle,where cosθ = -1/2.
In the unit circle, we have two angles at which cosθ is equal to -1/2:
θ = 2π/3 and θ = 4π/3.
Thus, the zeros of r are θ = 2π/3 and θ = 4π/3.
Therefore, the maximum value of |r| is 5, and the zeros of r are θ = 2π/3 and θ = 4π/3.