https://file.aiquickdraw.com/mathbot/file/1721929347373zo6w2arb.jpg

https://file.aiquickdraw.com/mathbot/file/1721929347373wrlsh7b2.jpg

Okay, just follow what the images say to do. FOR CHARLES,  be very simple in ur math. Anyways, for Francis since I can’t graph it for me unless u can on my graph, please ensure that you tell me the points to plot but anyways just make it VERY freaking simple because my teacher is going to grade this 😀😀😀.
Here is francoi’s skate info to graph but just anyway info do:
-François skate at the same speed as Charles during the first lap
- He slows down during the second lap and skates slower than Charles
- On the third lap, François skates even faster and wins the race
Write the time for each lap in the graph the Y axis is a distance from starting line in meters. The axis is the time in seconds. It starts from the origin and then it’s like the known points so five seconds 1015 20 25, 30, 35, 4045, 5060 you know that stuff And then for the distance for the wax intense so 10 20 30, 40, 5060 that type of thing

ONLY use garde 10 and 11 math. Keep it VERY ainple no crazy methods like it’s explained to a child in grade 3.. Solve using grade 11 functions and grade 10 msth. Only use methods from those grades and lower not higher than grade 11.
Show the solving steps and keep it VERY simplified like you’re. A teacher and you’re solving the problem. Include therefore statements underline what’s important from the q? Make it look human. I have to write this down. No sus pls

To solve this problem, we'll first determine the equation for Charles's skate from the provided graph, and then we'll graph François's skate based on the given information.

### Step 1: Determine the Equation for Charles's Skate

From the graph, we can see that Charles's skate follows a sinusoidal pattern. The distance from the starting line \( d(t) \) is a function of time \( t \). The graph shows that the distance varies between 0 and 50 meters, completing a cycle every 20 seconds.

- **Amplitude (A)**: The maximum distance from the equilibrium position (middle of the wave) is 25 meters.
- **Period (T)**: The time it takes to complete one full cycle is 20 seconds.
- **Frequency (f)**: The number of cycles per second. \( f = \frac{1}{T} = \frac{1}{20} \) cycles per second.
- **Vertical Shift (D)**: The middle of the wave is at 25 meters.
- **Phase Shift (C)**: The wave starts at the origin, so no phase shift is needed.

The general form of the sinusoidal function can be written as:
\[ d(t) = A \sin(B(t - C)) + D \]

Where:
\[ A = 25 \]
\[ B = \frac{2\pi}{T} = \frac{2\pi}{20} = \frac{\pi}{10} \]
\[ D = 25 \]

So the equation for Charles's skate is:
\[ \boxed{d(t) = 25 \sin\left(\frac{\pi}{10} t\right) + 25} \]

### Step 2: Determine Points for François's Skate

Using the given information about François's skate, we will plot his speed relative to Charles's speed on each lap.

1. **First Lap (0 to 20 seconds)**: François skates at the same speed as Charles.
   - Equation: \( d(t) = 25 \sin\left(\frac{\pi}{10} t\right) + 25 \)
   - Key Points: (0, 25), (10, 50), (20, 25)

2. **Second Lap (20 to 40 seconds)**: François skates slower than Charles.
   - Since François slows down, we can assume his amplitude is smaller. Let's assume \( \frac{2}{3} \) of Charles's amplitude.
   - New Amplitude: \( 25 \times \frac{2}{3} = 16.67 \)
   - Equation: \( d(t) = 16.67 \sin\left(\frac{\pi}{10} t - \frac{\pi}{2}\right) + 25 \)
   - Key Points: (20, 25), (30, 25), (40, 25) (midpoint; assume constant speed here)

3. **Third Lap (40 to 60 seconds)**: François skates faster than Charles.
   - Since François speeds up, we can assume his amplitude is larger. Let's assume \( \frac{4}{3} \) of Charles's amplitude.
   - New Amplitude: \( 25 \times \frac{4}{3} = 33.33 \)
   - Equation: \( d(t) = 33.33 \sin\left(\frac{\pi}{10} t\right) + 25 \)
   - Key Points: (40, 25), (50, 50), (60, 25)

### Graphing François's Skate

Now, let's plot the points for François's skate on the provided grid. The key points for each lap are:

1. **First Lap**:
   - (0, 25), (10, 50), (20, 25)
2. **Second Lap**:
   - (20, 25), (30, 25), (40, 25)
3. **Third

Learn how to graph François and Charles's skate race using sinusoidal functions. Solve step-by-step with grade 10 and 11 math methods. Includes plotting key points for each lap based on their speeds and distances.

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