Consider a circle of radius 2 and a point P rotating around it. The angle θ, in radians, is given as a function of time t.

(a) What is the value of θ when t=1.5?
(b) Find the coordinates of P when t=1.5

(a) Well, if θ is given as a function of time, we need to know the equation of that function to find the value of θ when t=1.5. Without that information, we might as well ask the circle to tell a joke!

Why don't scientists trust atoms?

Because they make up everything!

(b) Unfortunately, I can't directly provide the coordinates of point P without knowing the function that relates θ to time. But I can tell you that the coordinates of P will change as θ changes. So, let's have a little fun with math jokes instead!

Why was the math book sad?

Because it had too many problems!

Don't worry, though. With the right equation or more information, we can easily find the coordinates of P when t=1.5!

To answer these questions, we need to use some basic trigonometry and the properties of circles.

Given:
- A circle of radius 2.
- A point P rotating around the circle.
- The angle θ, in radians, is given as a function of time t.

(a) To find the value of θ when t=1.5, we need the specific function that relates t and θ. Once we have that function, we can substitute t=1.5 to find the corresponding value of θ.

(b) To find the coordinates of point P when t=1.5, we can use the relation between the angle θ and the coordinates (x, y) of the point P on the circle.

Let's assume that the center of the circle is at the origin (0, 0) of a coordinate system. Let's also assume that at time t=0, the point P is located at the positive x-axis (2, 0). As t increases, P rotates counterclockwise around the circle.

The equation for the perimeter of a circle, given the radius r and the angle θ (in radians), is P = 2πr(θ/2π) = rθ.

Considering our circle has a radius of 2, the equation for the perimeter becomes P = 4θ.

Let's now relate the angle θ to the coordinates (x, y) of point P using trigonometry.

Since P is rotating around the circle, we can express its coordinates using trigonometric functions. The x-coordinate can be found using the cosine function, and the y-coordinate can be found using the sine function.

x = r * cos(θ)
y = r * sin(θ)

Substituting the radius r=2, we get:
x = 2 * cos(θ)
y = 2 * sin(θ)

To find the coordinates of P when t=1.5, we need to determine the value of θ for that specific time and then use that value to find x and y.

Please provide the specific function that relates the angle θ to time t.

To find the value of θ when t=1.5, you need the function that relates θ to t. This function could be given or you may have to derive it from the given information. Once you have the function, substitute t=1.5 into the equation to calculate the value of θ.

To find the coordinates of point P when t=1.5, you need to use the trigonometric relationships in a circle. Since the point P is rotating around a circle of radius 2, the coordinates of P can be given by (x,y) = (r*cos(θ), r*sin(θ)), where r is the radius and θ is the angle.

Given that the radius is 2 and you have already obtained the value of θ when t=1.5, substitute it into the equation to find the x and y coordinates.

You did not say what sort of a function so I must leave it general.

Theta = f(1.5)

x = 2 cos (f(1.5))
y = 2 sin (f(1.5))