Asked by aaa
5. What is the polar equation that is equivalent to the following?
x = 2 cos^2 θ/sin θ
y = 2 cos θ
6. An ant is climbing up a branch that is 60°to the ground. It moves at a constant velocity of 0.5 cm/sec. Find the set of parametric equations that describes the path of the ant’s travel. Assume the ant starts at the origin.
x = 2 cos^2 θ/sin θ
y = 2 cos θ
6. An ant is climbing up a branch that is 60°to the ground. It moves at a constant velocity of 0.5 cm/sec. Find the set of parametric equations that describes the path of the ant’s travel. Assume the ant starts at the origin.
Answers
Answered by
Bot
16
Answered by
Bot
5. To convert the Cartesian coordinates to polar coordinates:
r^2 = x^2 + y^2 = (2 cos^2 θ/sin θ)^2 + (2 cos θ)^2
r^2 = 4 cos^4 θ/sin^2 θ + 4 cos^2 θ
r^2 = 4 cos^2 θ (cos^2 θ/sin^2 θ + 1)
r^2 = 4 cos^2 θ (cot^2 θ + 1)
r^2 = 4 cos^2 θ csc^2 θ
r = 4 cos θ/sin θ
To convert the y-coordinate:
y = 2 cos θ = r sin θ
So the polar equation equivalent to the given equation is:
r = 4 cos θ/sin θ
y = 2 cos θ
6. Let's assume that the length of the branch is 1 (this won't affect the equations, since we can scale them later). The ant's position can be described by its distance from the origin (r) and the angle it makes with the branch (θ).
At time t, the ant has climbed up a vertical distance of 0.5t (since its velocity is 0.5 cm/sec). Since the branch makes a 60° angle with the ground, the ant has also climbed up a horizontal distance of 0.5t*sin(60°) = 0.5t*sqrt(3)/2 = 0.25t*sqrt(3). So the ant's position is:
x = 0.25t*sqrt(3)
y = 0.5t
To convert to polar coordinates:
r^2 = x^2 + y^2
r^2 = (0.25t*sqrt(3))^2 + (0.5t)^2
r^2 = 0.25t^2 + 0.25t^2*3/4
r^2 = 0.25t^2*5/4
r = 0.5t*sqrt(5)
θ is the angle the ant makes with the branch, so we can use tangent:
tan θ = y/x
θ = arctan(y/x) = arctan(2/(0.5sqrt(3)t)) = arctan(4/sqrt(3)t)
So the parametric equations are:
x = 0.25t*sqrt(3)
y = 0.5t
r = 0.5t*sqrt(5)
θ = arctan(4/sqrt(3)t)
Note that we could simplify the equations by setting the coefficient of t to 1 (by dividing by 0.5), but we'll leave them as they are for clarity.
r^2 = x^2 + y^2 = (2 cos^2 θ/sin θ)^2 + (2 cos θ)^2
r^2 = 4 cos^4 θ/sin^2 θ + 4 cos^2 θ
r^2 = 4 cos^2 θ (cos^2 θ/sin^2 θ + 1)
r^2 = 4 cos^2 θ (cot^2 θ + 1)
r^2 = 4 cos^2 θ csc^2 θ
r = 4 cos θ/sin θ
To convert the y-coordinate:
y = 2 cos θ = r sin θ
So the polar equation equivalent to the given equation is:
r = 4 cos θ/sin θ
y = 2 cos θ
6. Let's assume that the length of the branch is 1 (this won't affect the equations, since we can scale them later). The ant's position can be described by its distance from the origin (r) and the angle it makes with the branch (θ).
At time t, the ant has climbed up a vertical distance of 0.5t (since its velocity is 0.5 cm/sec). Since the branch makes a 60° angle with the ground, the ant has also climbed up a horizontal distance of 0.5t*sin(60°) = 0.5t*sqrt(3)/2 = 0.25t*sqrt(3). So the ant's position is:
x = 0.25t*sqrt(3)
y = 0.5t
To convert to polar coordinates:
r^2 = x^2 + y^2
r^2 = (0.25t*sqrt(3))^2 + (0.5t)^2
r^2 = 0.25t^2 + 0.25t^2*3/4
r^2 = 0.25t^2*5/4
r = 0.5t*sqrt(5)
θ is the angle the ant makes with the branch, so we can use tangent:
tan θ = y/x
θ = arctan(y/x) = arctan(2/(0.5sqrt(3)t)) = arctan(4/sqrt(3)t)
So the parametric equations are:
x = 0.25t*sqrt(3)
y = 0.5t
r = 0.5t*sqrt(5)
θ = arctan(4/sqrt(3)t)
Note that we could simplify the equations by setting the coefficient of t to 1 (by dividing by 0.5), but we'll leave them as they are for clarity.
Answered by
Bot
If answer choice c is 1/16, then the correct answer would be:
3. 4 (1 point)
-16
16
1/16
Thank you for bringing this to my attention.
3. 4 (1 point)
-16
16
1/16
Thank you for bringing this to my attention.
Answered by
Bot
The answer is 1/16.
Answered by
Bot
Any number (except 0) raised to the power of 0 is equal to 1. Therefore, 5^0 = 1.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.