Match each polar equation below to the best description. Each answer should be C,F,I,L,M,O,or T.
DESCRIPTIONS
C. Cardioid, F. Rose with four petals, I. Inwardly spiraling spiral, L. Lemacon, M. Lemniscate, O. Outwardly spiraling spiral, T. Rose with three petals
1. r=10-10sin(theta) ?
2. r=10/theta r>0 ?
3. r=18sin(2theta) ?
4. r=10theta r>o ?
5. r=10+20cos(theta) ?
6. r=10cos(3theta) ?
To match each polar equation to the corresponding description, we need to understand the characteristics of each type of polar equation.
C. Cardioid: A cardioid is a polar equation that forms a heart-like shape. It usually has a single cusp and a symmetric shape.
F. Rose with four petals: A rose with four petals is a polar equation that creates a flower-like shape with four evenly spaced lobes.
I. Inwardly spiraling spiral: An inwardly spiraling spiral is a polar equation that forms a spiral that curves inward towards the pole.
L. Lemnicon: A lemniscate is a polar equation that generates a figure-eight shape.
M. Lemniscate: Similar to L, a lemniscate is a polar equation that creates a figure-eight shape.
O. Outwardly spiraling spiral: An outwardly spiraling spiral is a polar equation that forms a spiral that curves outward away from the pole.
T. Rose with three petals: A rose with three petals is a polar equation that creates a flower-like shape with three evenly spaced lobes.
Now, let's match each equation to the best description:
1. r = 10 - 10sin(theta): This equation forms a heart-like shape, which matches the description of a cardioid (C).
2. r = 10/theta, r > 0: This equation represents a curve that is not similar to any of the given descriptions.
3. r = 18sin(2theta): This equation generates a figure-eight shape, matching the description of a lemniscate (L or M).
4. r = 10theta, r > 0: This equation forms a spiral shape that curves inward, corresponding to the description of an inwardly spiraling spiral (I).
5. r = 10 + 20cos(theta): This equation creates a flower-like shape with four lobes, matching the description of a rose with four petals (F).
6. r = 10cos(3theta): This equation generates a flower-like shape with three evenly spaced lobes, matching the description of a rose with three petals (T).
So, the corresponding matches are:
1. C (Cardioid)
2. No Match
3. L or M (Lemniscate)
4. I (Inwardly spiraling spiral)
5. F (Rose with four petals)
6. T (Rose with three petals)