Question 1

A motor boat travels counterclockwise around a circular course with a radius of 4 miles, completing one lap every $$2\pi$$ hours at a constant speed. Which of the following parametric equations correctly describes the position of the boat at time $$t (in $$hours)?

Accepted Answer

$$x = 4 \cos t,\ y = 4 \sin t$$

Question 2

Given the parametric equations $$x = 2t$$, $$y = \frac{1}{2}t^2$ - 4$$ for $$0 \leq t \leq 8$$, what is the Cartesian equation relating $$y$$ and $$x$$?

Accepted Answer

$$y = \frac{x^2$}{8} - 4$$

Question 3

Which of the following best describes the positive orientation of a parametric curve?

Accepted Answer

The direction in which the curve is traced as the parameter increases.

Question 4

Suppose a turtle walks with constant speed in the counterclockwise direction on a circle of radius 4, completing one lap in 30 minutes. Which parametric equations describe the turtle's position at time $$t (in $$minutes)?

Accepted Answer

$$x = 4 \cos \left(\frac{\pi t}{15}\right),\ y = 4 \sin \left(\frac{\pi t}{15}\right)$$

Question 5

Given the parametric equations $$x = 4 \cos 2\pi t$$, $$y = 4 \sin 2\pi t$$ for $$0 \leq t \leq 1$$, what geometric figure do these equations describe?

Accepted Answer

A circle of radius 4 centered at the origin.

Question 6

Which of the following is the general parametric equation for a circle of radius $$a$$ centered at $$(x_0$, y_0$)$$, traced counterclockwise?

Accepted Answer

$$x = x_0$ + a \cos bt,\ y = y_0$ + a \sin bt$$

Question 7

Given the parametric equations $$x = -2 + 3t$$, $$y = 4 - 6t$$, for $$-\infty \u003c t \u003c \infty$$, what is the slope of the line described by these equations?

Accepted Answer

$$-2$$

Question 8

A line passes through the point $$(2,1)$$ with slope $$\frac{1}{3}. $$Which of the following is a valid pair of parametric equations for this line?

Accepted Answer

$$x = 2 + 3t,\ y = 1 + t$$

Question 9

Find the parametric equations for the line segment starting at $$P(4,7)$$ and ending at $$Q(2,-3). $$Which of the following is correct?

Accepted Answer

$$x = 4 - 1t,\ y = 7 - 5t$$

Question 10

Which of the following statements about parametric equations is TRUE?

Accepted Answer

They can describe curves that cannot be represented as functions $$y = f(x).$$

Question 11

If $$x = 4 \cos t$$ and $$y = 4 \sin t$$, what is the value of $$x^2$$ + $$y^2$$ for any $$t$$?

Accepted Answer

$$16$$

Question 12

A particle moves along a line so that its position at time $$t is $$given by $$x = x_0$ + at$$, $$y = y_0$ + bt. $$What is the slope of the line traced by the particle?

Accepted Answer

$$\frac{b}{a}$$

Question 13

Which of the following parametric equations describes a vertical line?

Accepted Answer

$$x = x_0$,\ y = y_0$ + bt$$

Question 14

If a circle is described by $$x = 3 + 5 \cos t$$, $$y = -2 + 5 \sin t$$, what is the center and radius of the circle?

Accepted Answer

Center $$(3, -2)$$, radius $$5$$

Question 15

A parametric curve is given by $$x = 4 \cos bt$$, $$y = 4 \sin bt. If $$the curve completes one revolution in $$T$$ units of time, what is the value of $$b$$?

Accepted Answer

$$b = \frac{2\pi}{T}$$

Parametric and Polar Curves: Foundations and Applications

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