Answer:
See below for answers and explanations
Step-by-step explanation:
Problem 1
- The equation for a rose curve in polar form is either [tex]r=a\:cos(n\theta)[/tex] if the pole is horizontal or [tex]r=a\:sin(n\theta)[/tex] if the pole is vertical.
- The value [tex]a[/tex] represents the length of each petal
- If [tex]n[/tex] is odd, there will be [tex]n[/tex] petals
- If [tex]n[/tex] is even, there will be [tex]2n[/tex] petals
Looking at the graph, the pole is vertical, which means that our polar curve will take the form of [tex]r=a\:sin(n\theta)[/tex]. We see that there are 12 petals, so this must mean that [tex]n=6[/tex]. Additionally, the length of each petal is 4 units, so [tex]a=4[/tex]. This means our equation is [tex]r=4\:sin6\theta[/tex], making B the correct answer.
Problem 2
This polar curve is a limaçon:
- A limaçon is in the form of either [tex]r=a\pm bsin\theta[/tex] or [tex]r=a\pm bcos\theta[/tex], depending on the direction of the pole where [tex]a > 0[/tex] and [tex]b > 0[/tex]
- If [tex]\frac{a}{b} < 1[/tex], the limaçon is an inner loop limaçon
- If [tex]\frac{a}{b}=1[/tex], the limaçon is a cardioid
- If [tex]1 < \frac{a}{b} < 2[/tex], the limaçon is dimpled with no inner loop
- If [tex]\frac{a}{b}\geq 2[/tex], the limaçon will be convex with no dimple and no inner loop
Given that our graph is a cardioid (heart-shaped) and it has a vertical pole, this means that our equation takes the form of [tex]r=a\pm bsin\theta[/tex] where [tex]\frac{a}{b}=1[/tex]. However, none of the answers look correct
Problem 3
Only [tex]r=4+7sin\theta[/tex] is correct since it takes the form of [tex]r=a\pm bsin\theta[/tex]
Problem 4
The loop starts at [tex]\theta=\frac{2\pi}{3}[/tex] if you look at this on a polar graph and ends at [tex]\theta=\pi[/tex], so this means that [tex]\frac{2\pi}{3}\leq\theta\leq \pi[/tex] is the correct answer
Problem 5
We could graph all the points, but the more important points to notice are [tex](0,4)[/tex], [tex](\frac{\pi}{2},0)[/tex], and [tex](\pi,-4)[/tex]. These show us the petals of the curve, which happens to be the option in the picture. This exact function is [tex]r=4\:cos2\theta[/tex]
