1.A wind turbine has blades 50m in diameter and an overall height of 125m. If it has four blades instead of three, create four equations modelling the height of a point on the tip for each of the four blades.
2.Create a sine equation for the height of a point as shown below.
Rotation Height
0° 140.0
30° 110.0
60° 88.0
90° 80.0
120° 88.0
150° 110.0
180° 140.0
210° 170.0
240° 192.0
270° 200.0
300° 192.0
330° 170.0
360° 140.0
3.A point on a wheel has an equation y = 10 sin (x - 45°) + 20 that models the height as the wheel rotates. Answer the following questions.
(a)What is the height if the wheel has rotated 135°?
(b)What are the possible values of rotation (i.e., the value of x) if the height is 15m?
(c)If the hub of the wheel were moved down 5m, what would the values of rotation (the values of x) be if the height were 15m?
a) 10 sin (135° - 45°) + 20 = 10 sin (90°) + 20 = 20 + 20 = 40m
b) x = 45° + 180° + n(360°), where n is an integer; x = 225° + n(360°)
c) x = 45° + 180° + n(360°) - 5, where n is an integer; x = 220° + n(360°)
Solution:
1. To create equations modeling the height of a point on each of the four blades, we can use the equation of a circle. Here are the equations:
For Blade 1:
h1(x) = (-d/2) * cos(θ) + H/2
For Blade 2:
h2(x) = (-d/2) * cos(θ + 90) + H/2
For Blade 3:
h3(x) = (-d/2) * cos(θ + 180) + H/2
For Blade 4:
h4(x) = (-d/2) * cos(θ + 270) + H/2
where d is the diameter of the blades (50m) and H is the overall height of the wind turbine (125m).
2. To create a sine equation for the given set of data, we can use the sine function in the general form: y = A * sin(Bx + C) + D. Here's the equation:
y = 30 * sin((π/180)x) + 110
where A = 30, B = π/180, C = 0, and D = 110.
3. For the given equation y = 10 * sin(x - 45°) + 20, let's answer the questions:
(a) To find the height when the wheel has rotated 135°, substitute x = 135° into the equation:
y = 10 * sin(135° - 45°) + 20
y = 10 * sin(90°) + 20
y = 10 + 20
y = 30
So, the height when the wheel has rotated 135° is 30m.
(b) To find the possible values of rotation (x) when the height is 15m, we can solve the equation:
15 = 10 * sin(x - 45°) + 20
-5 = 10 * sin(x - 45°)
sin(x - 45°) = -0.5
The possible values of x can be found using the inverse sine function:
x - 45° = arcsin(-0.5)
x - 45° = -30° or x - 45° = -150°
Solving for x, we get:
x = 15° or x = -105°
So, the possible values of rotation (x) when the height is 15m are 15° and -105°.
(c) If the hub of the wheel were moved down 5m, the new equation for the height would be y = 10 * sin(x - 45°) + 15. To find the values of rotation (x) when the height is 15m, we can use the same process as in part (b):
15 = 10 * sin(x - 45°) + 15
0 = 10 * sin(x - 45°)
sin(x - 45°) = 0
The possible values of x can be found using the inverse sine function:
x - 45° = arcsin(0)
x - 45° = 0°
Solving for x, we get:
x = 45°
So, the values of rotation (x) when the height is 15m and the hub is moved down 5m are 45°.
1. To model the height of a point on the tip of each of the four blades of a wind turbine, we can use the equation of a circle.
Let's assume the center of the wind turbine is at the origin (0,0). The radius of the circle formed by the blades is half of the diameter, which is 25m.
For the first blade, the equation of the circle can be represented as:
(x - 0)^2 + (y - 0)^2 = 25^2
Simplifying it, we get:
x^2 + y^2 = 625
Now, let's consider the different positions of the blades. If the blade is at an angle of 0°, the tip of the blade will be at the highest point. If it is at 90°, it will be at the lowest point, and so on.
For the four blades, we can use the following equations:
Blade 1: x^2 + y^2 = 625
Blade 2: (x - 50)^2 + y^2 = 625
Blade 3: x^2 + (y - 125)^2 = 625
Blade 4: (x - 50)^2 + (y - 125)^2 = 625
These equations model the height of a point on the tip of each of the four blades of the wind turbine.
2. To create a sine equation for the given data, we need to find the amplitude, period, phase shift, and vertical shift.
Looking at the data provided, we can observe that the maximum height is 140, and the minimum height is 80. So the amplitude is half the difference between the maximum and minimum values, which is (140 - 80) / 2 = 30.
The period of a sine function is the distance between two consecutive peaks or troughs. From the given data, the period is 360°, as the pattern repeats after every 360° of rotation.
To find the phase shift, we need to determine where the function starts. In this case, the function starts at 0°, so the phase shift is 0.
The vertical shift is the displacement of the graph above or below the x-axis. In this case, the average of the maximum and minimum heights is (140 + 80) / 2 = 110, so the vertical shift is 110.
Putting all these values together, the sine equation representing the height of a point as shown below is:
y = 30 sin(x) + 110
3. (a) To find the height when the wheel has rotated 135°, we can substitute x = 135° into the given equation:
y = 10 sin (x - 45°) + 20
y = 10 sin (135° - 45°) + 20
y = 10 sin 90° + 20
y = 10 + 20
y = 30
So, the height when the wheel has rotated 135° is 30m.
(b) To find the possible values of rotation (x) when the height is 15m, we can set y = 15 in the equation and solve for x:
15 = 10 sin (x - 45°) + 20
Subtracting 20 from both sides:
-5 = 10 sin (x - 45°)
Dividing by 10:
-0.5 = sin (x - 45°)
To find the possible values of x, we can take the inverse sine (sin^(-1)) of -0.5:
x - 45° = sin^(-1) (-0.5)
x - 45° = -30° or x - 45° = -150°
Adding 45° to both sides:
x = 15° or x = -105°
So, the possible values of rotation (x) when the height is 15m are 15° and -105°.
(c) If the hub of the wheel were moved down 5m, the equation representing the wheel's height would change to:
y = 10 sin (x - 45°) + 15
To find the new values of rotation (x) when the height is 15m, we can set y = 15 in the equation and solve for x:
15 = 10 sin (x - 45°) + 15
Subtracting 15 from both sides:
0 = 10 sin (x - 45°)
Dividing by 10:
0 = sin (x - 45°)
Since sin(0) = 0, the value of x - 45° could be any multiple of 360°.
So, the values of rotation (x) when the height is 15m and the hub is moved down 5m can be expressed as x = 45° + n * 360°, where n is an integer.