Question

Please help with all questions !!

Last question(#5) has one of the answer choices with it. I couldn't post the rest of the answer choices. Find what the points should be graphed on.

Answer 1

See below for answers and explanations

Explanation:

Problem 1

• The equation for a rose curve in polar form is either
  `r=a\:cos(n\theta)` if the pole is horizontal or
  `r=a\:sin(n\theta)` if the pole is vertical.
• The value
  `a` represents the length of each petal

• If
  `n` is odd, there will be
  `n` petals
• If
  `n` is even, there will be
  `2n` petals

Looking at the graph, the pole is vertical, which means that our polar curve will take the form of
`r=a\:sin(n\theta)`. We see that there are 12 petals, so this must mean that
`n=6`. Additionally, the length of each petal is 4 units, so
`a=4`. This means our equation is
`r=4\:sin6\theta`, making B the correct answer.

Problem 2

This polar curve is a limaçon:

• A limaçon is in the form of either
  `r=a\pm bsin\theta` or
  `r=a\pm bcos\theta`, depending on the direction of the pole where
  `a > 0` and
  `b > 0`
• If
  `(a)/(b) < 1`, the limaçon is an inner loop limaçon
• If
  `(a)/(b)=1`, the limaçon is a cardioid
• If
  `1 < (a)/(b) < 2`, the limaçon is dimpled with no inner loop
• If
  `(a)/(b)\geq 2`, 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
`r=a\pm bsin\theta` where
`(a)/(b)=1`. However, none of the answers look correct

Problem 3

Only
`r=4+7sin\theta` is correct since it takes the form of
`r=a\pm bsin\theta`

Problem 4

The loop starts at
`\theta=(2\pi)/(3)` if you look at this on a polar graph and ends at
`\theta=\pi`, so this means that
`(2\pi)/(3)\leq\theta\leq \pi` is the correct answer

Problem 5

We could graph all the points, but the more important points to notice are
`(0,4)`,
`((\pi)/(2),0)`, and
`(\pi,-4)`. These show us the petals of the curve, which happens to be the option in the picture. This exact function is
`r=4\:cos2\theta`