(Day 2) Unit 9 Lesson 7 Answer Key
1. It’s the bottom of the ninth inning, the Cubs are behind 6-3 and the bases are loaded.
Sammy Sosa is at bat. He swings and makes contact with the ball 3 feet above the plate at
an angle of 20 degrees from the horizontal at a velocity of 150 feet per second. He hits
straight toward center field where there is a fence 400 feet from home plate and 20 feet
high. Neglect resistance due to wind. Does Sammy hit a grand slam and win the game?
Sammy does not hit a grand slam. The ball is 0.26 feet short of going over the wall.
2. Chris and Linda warm up on the outfield by tossing softballs to each other. Suppose both
players tossed a ball at the same time at a height of 5 feet. Chris threw the ball with a
velocity of 45 ft/sec at a 44 degree angle and Linda threw the ball with a velocity of 41
ft/sec at a 39 degree angle. They are standing 78 feet apart. What is the minimum
distance between the two balls and at what time does the minimum distance occur?
Chris (graphed in blue)
Linda (graphed in red)
t
0
0.2
0.4
0.6
0.8
x(Chris)
0
6.47
12.95 19.42 25.9
y(Chris)
5
10.6
14.94 18
1
1.2
1.4
1.6
32.37 38.84 45.32 51.8
19.77 20.26 19.47 17.4
14
x(Linda) 78 71.63 65.25 58.88 52.5
46.14 39.76 33.4
27.02
y(Linda) 5
12.76 14.72 15.4
14.8
12.9
9.76
5.32
52.3
14.8
6.6
14.2
26.3
9.52
Distance 78 65.2
39.6
27
1.8
2
58.27 64.74
9.43
3.52
The minimum distance occurs at t = 1.2 seconds and the distance is 6.6 feet ….Create
a table and use distance formula.
3. Tony and Sue are launching yard darts 20 feet from the front edge of a circular target of a
radius of 18 inches. Tony throws the dart directly at the target and releases it 3 feet above
the ground with an initial velocity of 30 feet per second at a 70 degree angle. Will he hit
the target? Sue releases the dart 4 feet above the ground with an initial velocity of 25 feet
per second at a 55 degree angle. Will she hit the target?
Tony
Sue
Tony
The dart has already hit the ground, so Tony will not hit the target.
Sue
The radius of the target is 1.5 feet, which means the target has a diameter of 3 feet.
Since Sue’s dart is at a height in between 0 and 3 feet at a horizontal distance of 20
feet she will hit the target.
4. Orlando hits a ball when it is 4 feet above ground level with an initial velocity of 160 feet
per second. The ball leaves the bat at an angle of 20 degrees with the horizontal and heads
toward a 30 foot fence 400 feet from home plate. How strong must a wind gust be (in
feet per second) that acts directly with or against the ball in order for the ball to hit
within a few inches of the top of the wall?
The parametric equations are
The wind only affects the horizontal distance the ball travels, therefore the wind
velocity is only included with the horizontal parametric equation. The wind is + if it
is in the direction of the hit and the wind is - if it is against the ball.
...Use quadratic formula to find t at that height.
t = 2.85 and 0.57
Now use time and horizontal distance to solve for v.
v = -10 ft/sec
v = 551.4 ft/sec
Since 551 ft/sec is unrealistic the wind is blowing against the ball at 10 ft/sec.
5. Jane is riding a Ferris wheel with a radius of 30 feet. The Ferris wheel is turning
counterclockwise at the rate of one revolution every 10 minutes. Assume the lowest point
of the Ferris wheel is 10 feet above the ground. At what time is Jane at a height of 68.5
feet?
The parametric equations are
Use u substitution and let u =
u = 71.8 degrees and 108.2 degrees
6. Consider a line, m, with parametric equations x = 2 + 3t and y = –t + 5. Write a set of
parametric equations for the line n perpendicular to m containing the point (4, 10).
Need to find the rectangular question for line m
(
)
Line n is perpendicular therefore has a slope of 3 and has the point (4, 10)
Sample Answers:
x=t
y = 3t – 2
x=t+1
y = 3(t + 1) - 2
7. The graph below models the path of a soccer ball kicked by one player and then headed
back by another player. The path of the initial kick is shown with a solid curve, and the path
of the headed ball is shown with a dashed curve.
a) If the ball is initially kicked at an angle of 50 degrees, find the initial speed of the ball.
7
v = 27.3 feet/sec
b) At what time does the ball reach the second player if the second player is standing about
17.5 feet away?
t = 0.997 seconds
c) If the second player heads the ball at an angle of 75 degrees, an initial speed of 8 ft/sec,
and at a height of 4.75 feet, approximately how long does the ball stay in the air from the
time it is first kicked until it lands?
…Quadratic formula
t – 0.84 seconds + 0.997 = 1.83 seconds.