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 best description, we will analyze the characteristics of each equation.
1. r = 10 - 10sin(theta):
This polar equation represents a cardioid shape, which is a heart-like curve. The equation contains a term of sin(theta) that determines the shape.
2. r = 10 / theta (r > 0):
This polar equation represents an outwardly spiraling spiral. The variable theta appears in the denominator, causing the radius to decrease as theta increases, resulting in a spiral shape.
3. r = 18sin(2theta):
This polar equation represents a rose with four petals. The equation contains a term of sin(2theta) which results in two cycles of the sine function in a full rotation, creating four petals.
4. r = 10theta (r > 0):
This polar equation represents an inwardly spiraling spiral. The radius is directly proportional to the angle theta, causing the distance from the origin to increase as theta increases, resulting in a spiral shape.
5. r = 10 + 20cos(theta):
This polar equation represents an outwardly spiraling spiral. The equation includes a term of cos(theta) that determines the shape. The radius increases as theta increases, resulting in a spiral shape.
6. r = 10cos(3theta):
This polar equation represents a rose with three petals. The equation contains a term of cos(3theta) which results in three cycles of the cosine function in a full rotation, creating three petals.
Now, let's match each polar equation to the corresponding description:
1. r = 10 - 10sin(theta) --> C. Cardioid
2. r = 10 / theta (r > 0) --> O. Outwardly spiraling spiral
3. r = 18sin(2theta) --> F. Rose with four petals
4. r = 10theta (r > 0) --> I. Inwardly spiraling spiral
5. r = 10 + 20cos(theta) --> O. Outwardly spiraling spiral
6. r = 10cos(3theta) --> T. Rose with three petals