https://file.aiquickdraw.com/mathbot/file/1726128412868907b7t1d.pdf

The document contains several math problems, focusing primarily on harmonic motion and the behavior of a Ferris wheel gondola. Below, I’ll walk through how to approach each of these problems based on the provided information.

### **Opgave 1:**
You are given the general formula for harmonic oscillation:

\[
f(x) = A \cdot \sin(\omega \cdot x + \phi) + d
\]

**a) Bestem forskriften for \( f \) ved aflæsning**  
- You need to determine the values of \( A \), \( \omega \), \( \phi \), and \( d \) based on the graph provided (although I can't see the graph, you can solve this by identifying amplitude \( A \), frequency \( \omega \), phase shift \( \phi \), and vertical shift \( d \)).

### **Opgave 2:**
The equation of motion for a gondola on the "London Eye" Ferris wheel is:

\[
f(t) = 67.5 \cdot \sin(0.209t - 1.57) + 70
\]

where \( f(t) \) is the height above the ground (in meters) at time \( t \) (in minutes).

**a) Tegn grafen for \( f \):**  
- To draw the graph, plot the sine function with an amplitude of 67.5, a phase shift of \( -1.57 \), and a vertical shift of 70. The period of the sine wave can be determined from the coefficient \( 0.209 \).

**b) Bestem gondolens højde 7 minutter efter start:**  
To find the height after 7 minutes, substitute \( t = 7 \) into the equation:

\[
f(7) = 67.5 \cdot \sin(0.209 \cdot 7 - 1.57) + 70
\]

**c) Bestem de tidspunkter, hvor gondolen befinder sig 40 meter over jordoverfladen:**  
Solve the equation for \( t \) when \( f(t) = 40 \):

\[
67.5 \cdot \sin(0.209t - 1.57) + 70 = 40
\]

This will require solving a trigonometric equation for \( t \), using inverse sine and adjusting for the period of the sine wave.

### **Opgave 3–5:**  
These tasks aren't fully visible, but based on their placement in the document, they might involve similar mathematical models or other problems related to trigonometry or calculus.

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Would you like to go through the solution of a particular part, or do you need a more detailed explanation for solving one of these problems? Let me know how you'd like to proceed!

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**Questions:**
1. How do you identify amplitude and phase shift from a graph of a sine wave?
2. What is the general strategy for solving trigonometric equations like \( f(t) = 40 \)?
3. How do you calculate the period of a sine function based on its equation?
4. What steps are involved in graphing a sine function with vertical and horizontal shifts?
5. How does the value of \( \omega \) affect the frequency of oscillation in harmonic motion?

**Tip:**  
To solve trigonometric equations, always check for multiple solutions within the given interval due to the periodic nature of sine functions.

This page provides solutions for problems related to harmonic motion and Ferris wheel height calculations. Learn to derive formulas for harmonic oscillation and solve real-world scenarios using trigonometric functions. Suitable for high school students in grades 10-12.

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