Question
Question 3 (4 points)
If the ride lasts for 10 minutes when will Milton be at a height of 25m? Show work
If the ride lasts for 10 minutes when will Milton be at a height of 25m? Show work
Answers
GPT-4o mini
To determine when Milton will reach a height of 25 meters during a 10-minute ride, we need more context about the situation, such as the type of ride (e.g., a Ferris wheel, roller coaster, etc.), the motion function (linear, circular, sinusoidal, etc.), and the height formula.
For example, if we assume this is a Ferris wheel and its motion can be modeled by a sine or cosine function, we typically express height \( H(t) \) in terms of time \( t \).
Let's say the height of the Ferris wheel above ground can be modeled by the function:
\[
H(t) = A \cdot \sin(Bt + C) + D
\]
Where:
- \( A \) is the amplitude (the maximum height deviation from the center).
- \( B \) affects the period of the motion.
- \( C \) is a phase shift.
- \( D \) is the vertical shift (the height of the center of the Ferris wheel).
If we have specific values for \( A \), \( B \), \( C \), and \( D \) based on the ride’s specifications, we could then solve for \( t \) when \( H(t) = 25 \).
### Hypothetical Example:
- Let’s assume \( A = 10 \), \( D = 15 \), and the wheel completes one full revolution in 10 minutes:
- Thus, \( B = \frac{2\pi}{10} = \frac{\pi}{5} \).
The height function would be:
\[
H(t) = 10 \cdot \sin\left(\frac{\pi}{5} t\right) + 15
\]
To find when \( H(t) = 25 \):
\[
10 \cdot \sin\left(\frac{\pi}{5} t\right) + 15 = 25
\]
\[
10 \cdot \sin\left(\frac{\pi}{5} t\right) = 10
\]
\[
\sin\left(\frac{\pi}{5} t\right) = 1
\]
The sine function reaches 1 at:
\[
\frac{\pi}{5} t = \frac{\pi}{2} + 2k\pi \quad (k \in \mathbb{Z})
\]
For the first occurrence (\( k = 0 \)):
\[
\frac{\pi}{5} t = \frac{\pi}{2}
\]
\[
t = \frac{\pi/2}{\pi/5} = \frac{5}{2} = 2.5 \text{ minutes}
\]
Thus, Milton will be at a height of 25 meters at **2.5 minutes** into the ride.
If you have a different scenario or more specific details about the ride's function, please provide that information for a precise answer!
For example, if we assume this is a Ferris wheel and its motion can be modeled by a sine or cosine function, we typically express height \( H(t) \) in terms of time \( t \).
Let's say the height of the Ferris wheel above ground can be modeled by the function:
\[
H(t) = A \cdot \sin(Bt + C) + D
\]
Where:
- \( A \) is the amplitude (the maximum height deviation from the center).
- \( B \) affects the period of the motion.
- \( C \) is a phase shift.
- \( D \) is the vertical shift (the height of the center of the Ferris wheel).
If we have specific values for \( A \), \( B \), \( C \), and \( D \) based on the ride’s specifications, we could then solve for \( t \) when \( H(t) = 25 \).
### Hypothetical Example:
- Let’s assume \( A = 10 \), \( D = 15 \), and the wheel completes one full revolution in 10 minutes:
- Thus, \( B = \frac{2\pi}{10} = \frac{\pi}{5} \).
The height function would be:
\[
H(t) = 10 \cdot \sin\left(\frac{\pi}{5} t\right) + 15
\]
To find when \( H(t) = 25 \):
\[
10 \cdot \sin\left(\frac{\pi}{5} t\right) + 15 = 25
\]
\[
10 \cdot \sin\left(\frac{\pi}{5} t\right) = 10
\]
\[
\sin\left(\frac{\pi}{5} t\right) = 1
\]
The sine function reaches 1 at:
\[
\frac{\pi}{5} t = \frac{\pi}{2} + 2k\pi \quad (k \in \mathbb{Z})
\]
For the first occurrence (\( k = 0 \)):
\[
\frac{\pi}{5} t = \frac{\pi}{2}
\]
\[
t = \frac{\pi/2}{\pi/5} = \frac{5}{2} = 2.5 \text{ minutes}
\]
Thus, Milton will be at a height of 25 meters at **2.5 minutes** into the ride.
If you have a different scenario or more specific details about the ride's function, please provide that information for a precise answer!