Answer :
Let's break down the problem and determine the correct sine model for the height of the windmill blade:
1. Understand the Cycle:
- The windmill completes 3 full rotations every minute. Since there are 60 seconds in a minute, each single rotation takes [tex]\( \frac{60}{3} = 20 \)[/tex] seconds.
2. Identify the Parameters:
- We are using the sine function of the form [tex]\( y = a \sin(b t) + k \)[/tex], where:
- [tex]\( a \)[/tex] is the amplitude, which corresponds to the length of the blade. In this case, [tex]\( a = 15 \)[/tex] feet.
- [tex]\( k \)[/tex] is the vertical shift, which is the height of the center of rotation above the ground. Here, [tex]\( k = 40 \)[/tex] feet.
- [tex]\( b \)[/tex] determines how quickly the sine function completes a cycle, given by the formula for the period [tex]\( T = \frac{2\pi}{b} \)[/tex].
3. Calculate [tex]\( b \)[/tex]:
- Since the windmill makes a complete rotation every 20 seconds, the period [tex]\( T = 20 \)[/tex].
- Using the period formula [tex]\( T = \frac{2\pi}{b} \)[/tex], we have:
[tex]\[
20 = \frac{2\pi}{b}
\][/tex]
- Solving for [tex]\( b \)[/tex], we get:
[tex]\[
b = \frac{2\pi}{20} = \frac{\pi}{10}
\][/tex]
4. Choose the Correct Sine Model:
- Now substituting the values we know into the sine model equation:
- [tex]\( a = 15 \)[/tex],
- [tex]\( b = \frac{\pi}{10} \)[/tex],
- [tex]\( k = 40 \)[/tex].
- The equation becomes:
[tex]\[
y = 15 \sin\left(\frac{\pi}{10} t\right) + 40
\][/tex]
Therefore, the correct sine model for the height of the end of the windmill blade as a function of time [tex]\( t \)[/tex] in seconds is:
[tex]\[ y = 15 \sin\left(\frac{\pi}{10} t\right) + 40 \][/tex]
1. Understand the Cycle:
- The windmill completes 3 full rotations every minute. Since there are 60 seconds in a minute, each single rotation takes [tex]\( \frac{60}{3} = 20 \)[/tex] seconds.
2. Identify the Parameters:
- We are using the sine function of the form [tex]\( y = a \sin(b t) + k \)[/tex], where:
- [tex]\( a \)[/tex] is the amplitude, which corresponds to the length of the blade. In this case, [tex]\( a = 15 \)[/tex] feet.
- [tex]\( k \)[/tex] is the vertical shift, which is the height of the center of rotation above the ground. Here, [tex]\( k = 40 \)[/tex] feet.
- [tex]\( b \)[/tex] determines how quickly the sine function completes a cycle, given by the formula for the period [tex]\( T = \frac{2\pi}{b} \)[/tex].
3. Calculate [tex]\( b \)[/tex]:
- Since the windmill makes a complete rotation every 20 seconds, the period [tex]\( T = 20 \)[/tex].
- Using the period formula [tex]\( T = \frac{2\pi}{b} \)[/tex], we have:
[tex]\[
20 = \frac{2\pi}{b}
\][/tex]
- Solving for [tex]\( b \)[/tex], we get:
[tex]\[
b = \frac{2\pi}{20} = \frac{\pi}{10}
\][/tex]
4. Choose the Correct Sine Model:
- Now substituting the values we know into the sine model equation:
- [tex]\( a = 15 \)[/tex],
- [tex]\( b = \frac{\pi}{10} \)[/tex],
- [tex]\( k = 40 \)[/tex].
- The equation becomes:
[tex]\[
y = 15 \sin\left(\frac{\pi}{10} t\right) + 40
\][/tex]
Therefore, the correct sine model for the height of the end of the windmill blade as a function of time [tex]\( t \)[/tex] in seconds is:
[tex]\[ y = 15 \sin\left(\frac{\pi}{10} t\right) + 40 \][/tex]
