To find the maximum [tex]\( r \)[/tex]-value for the polar equation [tex]\( r = -3 + 5 \sin \theta \)[/tex], we need to analyze the behavior of the function.
1. Understand the equation: The given polar equation is [tex]\( r = -3 + 5 \sin \theta \)[/tex]. Here, [tex]\( \theta \)[/tex] is the variable that affects the value of [tex]\( \sin \theta \)[/tex].
2. Range of sine function: The sine function, [tex]\( \sin \theta \)[/tex], oscillates between -1 and 1 for all values of [tex]\( \theta \)[/tex]. This means:
[tex]\[
-1 \leq \sin \theta \leq 1
\][/tex]
3. Determine the extreme values for [tex]\( r \)[/tex]:
- When [tex]\( \sin \theta \)[/tex] is at its minimum value, [tex]\(-1\)[/tex]:
[tex]\[
r = -3 + 5(-1) = -3 - 5 = -8
\][/tex]
- When [tex]\( \sin \theta \)[/tex] is at its maximum value, [tex]\(1\)[/tex]:
[tex]\[
r = -3 + 5(1) = -3 + 5 = 2
\][/tex]
4. Finding the maximum [tex]\( r \)[/tex]-value:
We need to find the maximum value of [tex]\( r = -3 + 5 \sin \theta \)[/tex]. From the calculations in step 3:
- The minimum value of [tex]\( r \)[/tex] is [tex]\(-8\)[/tex]
- The maximum value of [tex]\( r \)[/tex] is [tex]\( 2 \)[/tex]
5. Conclusion:
The maximum [tex]\( r \)[/tex]-value for the polar equation [tex]\( r = -3 + 5 \sin \theta \)[/tex] is [tex]\( 2 \)[/tex].
Among the given choices:
a. 8
b. 5
c. 2
d. 3
The correct answer is [tex]\(\boxed{2}\)[/tex].
