# 10.3 OFSA

## OFSA for Parametrics

This page contains peer generated questions that help you assess your understanding. Remember that questions may not represent the rigor of the questions you may be expected to complete on formal assessments. This page is simply an OFSA - opportunity for self assessment.

Question 1
Given
$$x=3sint$$
and
$$y=5cost$$
What is the shape of the x-y graph and how does the particle travel as time goes on?

a. ellipse, counterclockwise
b. line, oscillates- up then down
c. ellipse, clockwise
d. line, oscillates- down then up

Question 2

Two ships, the USS Lizzy and the USS Izzy are headed toward each other on path defined by:

USS Lizzy:
 [x=t]

and
 [y=t+30]

USS Izzy:
 [x=80-t]

and
 [y=t+10]

When will the two paths cross, but without colliding?

a. never
b. at (30,60) when USS Lizzy's t= 50
c. at (60,90) when USS Izzy's t= 30
d. at (30, 60) when USS Izzy's t= 50

Question 3
Given:
$$x=\frac{2}{t-3}$$

and

$$y=\frac{1}{t+5}$$

eliminate the parameter, and choose the correct graph.

a.

b.

c.

d.

Question 4
A soccer ball is traveling across the ice and is parametrically described by: x=2t+1and y=t-4. Ali is running across the field in hopes of intercepting the ball. The position of the tip of Ali's cleat is parametrically defined by: x=3t-10 and y= -t+3
Does Ali intercept the soccer ball? If yes, when?

a. yes, at 3.5 seconds
b. yes, at 11 seconds
c. yes, at 7.5 seconds
d. no, her foot and the ball do not collide

Question 5

Question 6
A football is thrown from a rooftop, 53 feet above the ground with an initial velocity of 26 ft/s and at an initial angle of 33° above the horizontal.
Create parametric equations to model the position of the ball in terms of the time 't' to find the horizontal distance to ball will travel before hitting the ground. (Calc. allowed!)

a.11.6 feet
b. 6.95 feet
c. 36.75 feet
d. 50.59 feet

Question 7- Simultaneous Solutions
$\\ \text{Find the values of x and y at each simultaneous intersection along with the value of t. Use radians.} \\ t \in \left[0, \pi\right] \\ x_1 = 10\sin(6t), \; y_1 = 4\cos(4t) + 3\\ x_2 = 10\cos(6t), \; y_2 = 3\cos(4t) + 3\\ \\ \text{A) no solutions } \\ \text{B) one solution: } (5\sqrt{2}, 3) \text{ at } t=\frac{3\pi}{8}} \\ \text{C) two solutions: } (5\sqrt{2}, 3) \text{ at } t=\frac{3\pi}{8} \text{ and } (-5\sqrt{2}, 3) \text{ at } t=\frac{7\pi}{8}} \\ \text{D) correct answer not give}$

Question 8
Tom and Jerry are at it again! Jerry is trying to cross a square room, which has a diagonal from (0,0) to (100,100). Tom's motion is described by the following equations: x = t + 17 and y = 5t^2 - 2t. Jerry's movement is described by: x = 3t - 1and y = 36t + 63. Will Jerry be captured by Tom? If so, find the x and y coordinates at the intersection.

$\\ \text{a) Yes, at (26, 27) when t = 9}\\ \text{b) Yes, at (26, 387) when t = 9} \\ \text{c) No} \\ \text{d) Yes, at (387, 26) when t = 9}$

Question 9

Question 10
A bug is crawling across the floor and its motion is described by x = 3t - 4 , y = 2t - 7. A man is walking across the same floor and his motion is described by x = 2t + 5 , y = t + 2. If they continue in this motion, will the man step on the bug?

a) No, he will not step on the bug
b) Yes, at (-13, -13)
c) Yes, at (9, 9)
d) Yes, at (23, 11)

Question 11
An object's motion is represented by x = 7 , y = (t + 2)(t - 4)^2. Describe the motion of the object over the interval -2 < t < 5

a) Travels vertically upward from (7, 0) to (7, 32), back down to (7, 0), and back up to (7, 7)
b) Travels horizontally right from (0, 7) to (32, 7), left to (0, 7), and right to (7, 7)
c) Travels vertically from (7, 0) to (7,7)
d) Travels vertically for a total displacement of 71 units

Question 12
What is the total distance traveled by an object oscillating between (1, 1) and (5, 7) in the interval 0 < t < 8? The object starts at (1, 1) and first reaches (5, 7) at
t = 4.

a) $\small \bg_white \fn_cm 2\sqrt{13}$
b) 0
c) $\small \bg_white \fn_cm 4\sqrt{13}$
d) 2

Question 13
Which description of motion below is not supported by the given parametric equations?
$\left\{\begin{matrix} x(t) = 2 \\ y(t)=(t-1)(t+2)^2 \end{matrix}\right. ; -5 \leq t \leq 5 \\ \\ \text{A) increases in y-coord to 0} \\ \text{B) increases in y-coord to 149} \\ \text{C) decreases in y-coord to -4} \\ \text{D) increases in y-coord to -4} \\$

Question 14

Robbie Gould, a place kicker for the Chicago Bears, kicks a football in a game with the Detroit Lions. The ball leaves the ground with a velocity of 89 ft./s at an angle of 63 degrees above the horizontal. If the ball is kicked from the Bears' 30 yard line and is aimed straight down the field, where does it land?

a) Between the 1 and 2 yard line.
b) Between the 3 and 4 yard line.
c) Between 4 and 5 yard line.
d) Off the field!

Question 15

Write the parametric equations, t-x and t-y, that produce this rectangular graph.

A) x=3sin(π/6t), y= -3cos((π/6t)
B) x= 3sin(π/8t), y= 3cos(π/8t)
C) x= 3sin(π/8t), y= -3cos(π/8t)
D) x=-3cos(π/8t), y=3sin(π/8t)

Question 16
A ball is thrown from a roof that is 54 feet above ground with an initial velocity of 40 ft/s at an angle 30 degrees above horizontal. How far will the ball travel horizontally before it hits the ground?

a)
b)
c)
d)

Question 17
Find the simultaneous solution for the following:
{x = 5t -6
{y= t^2 + 1

{x= 2t + 9
{y= T^2 + 1

a) (19,26) when t = 5
b) ( 26, 19) when t= 5
c) (-36, 37) when t= -6
d) (43.5, 57. 25)

Question 18
For t values [0, 5], a particle travels along the following path: y= ( t- 5)( t- 1)^2
What is the total displacement and total distance traveled by the particle?

a) Displacement 5, Distance 25
b) Displacement 0, Distance 20
c) Displacement 25, Distance 5
d) Displacement 5, Distance 5

Question 19 USE A CALCULATOR
Determine the simultaneous solution for the following equations:
x= 3 sin (πt)
y= 3 cos(πt) + 2

x= t- 2
y= t + 3

a) there is no simultaneous solution
b) cannot be determined
c) (0,5)
d) (-3, 2)

Question 20
Given the parametric equations below, which direction will the particle be traveling? (assume t: all real numbers)
$\left\{\begin{matrix}x=3cos(\frac{\Pi }{6}t) \\ y=-sin(\frac{\Pi }{6}t) \end{matrix}\right. \\ \\ \text{A) line; oscillating left to right} \\ \text{B) line; oscillating right to left} \\ \text{C) ellipse; CW} \\ \text{D) ellipse; CCW} \\$

$\left\{\begin{matrix}x=\sqrt{t} \\ y=2t+3 \end{matrix}\right.$
$\left\{\begin{matrix}x=\sqrt{6-t} \\ y=t^2 \end{matrix}$
$\\ A) \ t=3, (\sqrt3,9) \\ B) \ (i, 1) \\ C) \ (3.32,25) \\ D) \ \text{There are no non-simultaneous solutions.}$