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.
What is the height if the wheel has rotated 135°?
What are the possible values of rotation (i.e., the value of x) if the height is 15m?
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?
For your first question, simply let x = 135, set your calculator to DEG
and let your device do the work for you. Notice 135-45 = 90° and you should know sin 90° without a calculator.
I will do the 2nd question:
if y = 15
15 = 10sin(x-45) + 20
-1/2 = sin(x-45)
from basic trig you know sin 210° = -1/2 or sin 330° = -1/2
so x-45 = 210 or x-45 = 330
x = 255°, or x = 375°
there are other answer: since the period of sin(x-45°) is 360°
more answers can be obtained by adding or subtracting 360 to any answer
other answers:
we have 375-360 = 15° <----- that would be our smallest positive answers.
3rd part:
if moved down 5 units, our new equation is y = 10sin(x-45°) + 15
so now
15 = 10sin(x-45) + 15
you try it.
Oh, rotations and heights, we're getting all mathematical here! Let me put on my thinking cap and entertain you with some answers.
First, if the wheel has rotated by 135°, let's substitute that into the equation and find the height:
y = 10 sin (x - 45°) + 20
When x = 135°, we have:
y = 10 sin (135° - 45°) + 20
y = 10 sin (90°) + 20
y = 10 (1) + 20
y = 10 + 20
y = 30
So, the height would be 30m when the wheel has rotated 135°.
Now, let's move on to the second question. We are given that the height is 15m, and we need to find the possible values of rotation (x). Let's set up 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°)
Hmm, finding the angle when sin(x - 45°) is -0.5 is no laughing matter. To find possible values of x, we'll have to use our trusty calculator or some trigonometric wizardry.
Lastly, let's bring down the hub and see what happens. We are given that the height is 15m and the hub is moved down 5m. We have to find the values of rotation (x). Let's set up the equation and find x:
15 - 5 = 10 sin (x - 45°)
10 = 10 sin (x - 45°)
sine of (x - 45°) is still playing hard to get, but we'll keep trying!
I hope my humor managed to lighten up this mathematical journey. Remember, if all else fails, a good laugh can solve almost anything!
To find the height of the wheel if it has rotated 135°, we can substitute x = 135° into the equation y = 10 sin (x - 45°) + 20.
Step 1: Convert 135° to radians.
To convert degrees to radians, multiply by π/180.
135° * π/180 = 3π/4 radians.
Step 2: Substitute x = 3π/4 into the equation.
y = 10 sin ((3π/4) - 45°) + 20.
Step 3: Simplify the equation.
y = 10 sin (3π/4 - 45°) + 20.
y = 10 sin (3π/4 - 45°) + 20.
y = 10 sin (3π/4 - (45° * π/180)) + 20.
y = 10 sin (3π/4 - π/4) + 20.
y = 10 sin (2π/4) + 20.
y = 10 sin (π/2) + 20.
y = 10 * 1 + 20.
y = 10 + 20 = 30.
Therefore, the height of the wheel when it has rotated 135° is 30 meters.
To find the possible values of rotation (x) if the height is 15m, we need to rearrange the equation to solve for x.
Step 1: Subtract 20 from both sides of the equation.
y - 20 = 10 sin (x - 45°).
Step 2: Divide both sides of the equation by 10.
(y - 20)/10 = sin (x - 45°).
Step 3: Take the inverse sine (sin^-1) of both sides of the equation.
x - 45° = sin^-1 ((y - 20)/10).
Step 4: Add 45° to both sides of the equation.
x = sin^-1 ((y - 20)/10) + 45°.
Now we can substitute y = 15 into the equation to find the possible values of rotation (x) when the height is 15 meters.
x = sin^-1 ((15 - 20)/10) + 45°.
x = sin^-1 (-5/10) + 45°.
x = sin^-1 (-0.5) + 45°.
x ≈ -30° + 45° = 15°.
Therefore, the possible value of rotation (x) when the height is 15 meters is 15°.
If the hub of the wheel were moved down 5m, we need to adjust the equation by subtracting 5 from the y-coordinate.
New equation: y = 10 sin (x - 45°) + 20 - 5 = 10 sin (x - 45°) + 15.
To find the values of rotation (x) when the height is 15m, we can use the same process as before.
x = sin^-1 ((15 - 15)/10) + 45°.
x = sin^-1 (0) + 45°.
x = 0° + 45° = 45°.
Therefore, the value of rotation (x) when the height is 15m is 45°.
To find the height at a specific rotation angle, we can substitute the angle into the equation y = 10 sin(x - 45°) + 20. Let's solve each question step by step:
1. What is the height if the wheel has rotated 135°?
To find the height when the wheel has rotated 135°, we substitute x = 135° into the equation y = 10 sin(x - 45°) + 20:
y = 10 sin(135° - 45°) + 20
= 10 sin(90°) + 20
The sine of 90° is 1, so the equation simplifies to:
y = 10 * 1 + 20
y = 10 + 20
y = 30
Therefore, if the wheel has rotated 135°, the height is 30m.
2. What are the possible values of rotation (x) if the height is 15m?
To find the possible values of x when the height is 15m, we need to rearrange the equation:
15 = 10 sin(x - 45°) + 20
Subtracting 20 from both sides gives:
-5 = 10 sin(x - 45°)
Dividing both sides by 10 yields:
-0.5 = sin(x - 45°)
To find the angle whose sine is -0.5, we can use the arcsine function (sin⁻¹) or inverse sine. So we have:
x - 45° = sin⁻¹(-0.5)
Using a calculator or table of inverse trigonometric functions, we find that sin⁻¹(-0.5) is equal to -30° or -150°.
Adding 45° to both angles gives the possible values of x:
x = -30° + 45° = 15°
x = -150° + 45° = -105°
Therefore, when the height is 15m, the possible values of rotation (x) are 15° and -105°.
3. If the hub of the wheel were moved down 5m, what would be the values of rotation (x) if the height were 15m?
If the hub of the wheel is moved down 5m, the equation changes to: y = 10 sin(x - 45°) + 15. We want to find the values of x when the height is 15m. Let's set up the equation:
15 = 10 sin(x - 45°) + 15
Subtracting 15 from both sides gives:
0 = 10 sin(x - 45°)
Dividing both sides by 10 yields:
0 = sin(x - 45°)
This means the sine of (x - 45°) is 0. To find the angle whose sine is 0, we know it must be 0° or any multiple of 180°.
So we have the following possible values for (x - 45°):
x - 45° = 0° + 180°k, where k is an integer.
Solving for x:
x = 45° + 180°k, where k is an integer.
Therefore, if the height were 15m and the hub moved down 5m, the values of rotation (x) are 45° + 180°k, where k is an integer.