Find the maximum value of |r| and any zeros of r. (Assume 0 ≤ θ < 2π. Enter any zeros as a comma-separated list.)
r = 19 - 19sinθ
this is just a cardioid. maximum distance is 18 at θ=3π/2
to check,
r' = 19cosθ
r'=0 at θ = π/2, 3π/2
r is minimum at π/2, max at 3π/2
To find the maximum value of |r| and any zeros of r, we will first need to rewrite r in terms of trigonometric functions. We have r = 19 - 19sinθ.
The absolute value of r, denoted by |r|, is equal to the positive value of r when r is positive and the negative value of r when r is negative. Since r = 19 - 19sinθ can be positive or negative depending on the value of sinθ, we need to consider both cases.
Case 1: When r is positive
When r is positive, we have 19 - 19sinθ > 0. Rearranging the inequality, we get sinθ < 1. Since sinθ reaches its maximum value of 1 at θ = π/2, we have sinθ < 1 for all values of θ in the interval 0 ≤ θ < π/2. Therefore, r = 19 - 19sinθ is positive.
Case 2: When r is negative
When r is negative, we have 19 - 19sinθ < 0. Rearranging the inequality, we get sinθ > 1. However, sinθ can never be greater than 1 within the range 0 ≤ θ < 2π. Therefore, r = 19 - 19sinθ is always positive and has no zeros.
To find the maximum value of |r|, we can ignore the absolute value since r is always positive. Thus, the maximum value of r is 19.
In summary, the maximum value of |r| is 19 and r has no zeros within the given range.