The Flying Zucchini Circus Troupe has a human cannon-
ball act, shooting a performer from a cannon into a spe-
cially padded seat of a turning Ferris wheel. The Ferris
wheel has a radius of 40 feet and rotates counterclockwise at
one revolution per minute. The special seat starts at ground
level. Carefully explain why parametric equations for the
seat are
�
x
=
40 cos(
π
30
t
−
π
2
)
y
=
40
+
40 sin(
π
30
t
−
π
2
)
. The cannon is located
200 feet left of the Ferris wheel with the muzzle 10 feet
above ground. The performer is launched 35 seconds after
the wheel starts turning with an initial velocity of 100 ft/s
at an angle of
π
5
above the horizontal. Carefully explain
why parametric equations for the human cannonball are
�
x
=
(100 cos
π
5
)(
t
−
35)
−
200
y
=−
16(
t
−
35)
2
+
(100 sin
π
5
)(
t
−
35)
+
10
(
t
≥
35)
.
Determine whether the act is safe or the Flying Zucchini
comes down squash.
you really need to work on your text formatting.
well its not me who typed it shes someone else
To understand why the parametric equations for the seat and the human cannonball are as given, let's break down the problem step by step:
1. Parametric equations for the seat:
The seat of the Ferris wheel rotates counterclockwise at one revolution per minute. This means that the angle of rotation of the seat changes with time. We can use parametric equations to describe the position of the seat at any given time.
The x-coordinate of the seat can be obtained by considering the horizontal displacement from the center of the Ferris wheel. The radius of the Ferris wheel is 40 feet, and we know that the seat starts at ground level. To get the x-coordinate of the seat, we use the formula: x = r * cos(θ).
Here, θ represents the angle of rotation of the seat at any given time t. Since the seat starts at ground level, we need to adjust the initial angle. In this case, we subtract π/2 from the angle to ensure the seat starts at the correct position.
Therefore, the parametric equation for the x-coordinate of the seat is x = 40 * cos((π/30)t - π/2).
Similarly, the y-coordinate of the seat can be obtained by considering the vertical displacement from the center of the Ferris wheel. Since the seat starts at ground level, the y-coordinate is always constant and equal to the radius of the Ferris wheel, which is 40 feet.
Therefore, the parametric equation for the y-coordinate of the seat is y = 40.
2. Parametric equations for the human cannonball:
The cannon is located 200 feet left of the Ferris wheel, and the muzzle is 10 feet above the ground. The performer is launched 35 seconds after the wheel starts turning with an initial velocity of 100 ft/s at an angle of π/5 above the horizontal.
The x-coordinate of the human cannonball can be obtained by considering the horizontal displacement from the initial position of the cannon. We can use the formula: x = (initial velocity * cos(angle)) * (time - initial time) - initial x-coordinate.
In this case, the initial velocity is 100 ft/s and the angle is π/5. The initial time is 35 seconds, and the initial x-coordinate is 200 feet left of the Ferris wheel.
Therefore, the parametric equation for the x-coordinate of the human cannonball is x = (100 cos(π/5))(t - 35) - 200.
The y-coordinate of the human cannonball can be obtained by considering the vertical displacement from the initial position of the cannon. We can use the formula: y = (initial velocity * sin(angle)) * (time - initial time) + initial y-coordinate - (1/2 * acceleration * (time - initial time)^2).
In this case, the initial velocity is 100 ft/s, the angle is π/5, the initial time is 35 seconds, the initial y-coordinate is 10 feet above the ground, and the acceleration due to gravity is -32 ft/s^2 (negative because it acts downward).
Therefore, the parametric equation for the y-coordinate of the human cannonball is y = -16(t - 35)^2 + (100 sin(π/5))(t - 35) + 10.
Now, to determine whether the act is safe or not, we need to analyze the y-coordinate of the human cannonball. If the y-coordinate is positive (above the ground), then the performer lands safely on the seat. If the y-coordinate is negative (below the ground), then the performer does not land on the seat and the act is considered unsafe.
By analyzing the equation y = -16(t - 35)^2 + (100 sin(π/5))(t - 35) + 10, we can plug in different values of t after 35 seconds to determine the y-coordinate of the human cannonball.