<span>The particle starts at (3,5) and travels counter clockwise along a semi-circle with a radius of 2 and centered at (3,3) through the point (1,3) and stops at point (3,1).
First, let's look at the two equations:
x = 3 + 2 cos(t)
y = 3 + 2 sin(t)
If you ignore the limits specified for t, you'll realize that they describe a circle with a radius of 2, centered at (3, 3). Now that we know that the functions describe a circle, let's look at the limits specified for t and see what they imply.
Since the value is from π/2 to 3π/2, it will span a distance of π radians, so we know the particle will trace half of a full circle, or a semi-circle. Since the starting point is π/2, the starting coordinate of the particle will be:
(3 + 2 cos(Ď€/2), 3 + 2 sin(Ď€/2))
= (3 + 2*0, 3 + 2*1)
= (3+0, 3+2)
= (3,5)
And the ending point of the particle will obviously be (3,1). But to demonstrate:
(3 + 2 cos(3Ď€/2), 3 + 2 sin(3Ď€/2))
= (3 + 2*0, 3 + 2*(-1))
= (3+0, 3+ (-2))
= (3,1)
Now let's see where the particle is in the middle.
(3 + 2 cos(Ď€), 3 + 2 sin(Ď€))
= (3 + 2*(-1), 3 + 2*0)
= (3 + (-2), 3 + 0)
= (1, 3)
So the final description is:
The particle starts at (3,5) and travels counter clockwise along a semi-circle with a radius of 2 and centered at (3,3) through the point (1,3) and stops at point (3,1).</span>
I think this is asking what power when the first number given is raised to it will equal the number within the parentheses. So, for example, the answer to the first question would be 5 because 2^5 = 32. 10^3 = 1000, so the answer is 3, and so on.
