Let’s solve the problems one at a time from the image you’ve uploaded.

Problem 9

The function for blood pressure is given by: $P(t) = -20 \cos \left( \frac{5\pi}{3} t \right) + 100$

(a) What is the period of the function? What does the period represent?

The formula for the period of a cosine function is: $T = \frac{2\pi}{B}$ Here, $B = \frac{5\pi}{3}$. Substituting: $T = \frac{2\pi}{\frac{5\pi}{3}} = \frac{2\pi \cdot 3}{5\pi} = \frac{6}{5} \, \text{seconds}$

The period represents the time it takes for one complete cycle of blood pressure variation.

(b) How many times does the person’s heart beat each minute?

The number of heartbeats per minute is the reciprocal of the period (in seconds), scaled to 60 seconds: $\text{Beats per minute} = \frac{60}{T} = \frac{60}{\frac{6}{5}} = 60 \cdot \frac{5}{6} = 50 \, \text{beats per minute}$

(c) Sketch the graph of $P(t)$ for $0 \leq t \leq 6$.

I will sketch this for you based on the function. The graph will have:

• Amplitude: $20$
• Period: $T = 6/5$
• Vertical shift: $100$

(d) What is the range of the function? Explain its meaning.

The range of a cosine function is: $[\text{Vertical Shift} - \text{Amplitude}, \text{Vertical Shift} + \text{Amplitude}]$ Here, the vertical shift is $100$ and the amplitude is $20$, so: [ \text{Range} = [100 - 20, 100 + 20] = [80, 120] ]

This means the blood pressure oscillates between $80 \, \text{mmHg}$ (diastolic pressure) and $120 \, \text{mmHg}$ (systolic pressure).

Problem 10

A pendulum swings back and forth $10$ times in $8$ seconds, and the total horizontal distance is $40 \, \text{cm}$.

(a) Sketch the graph of this motion for two cycles.

The pendulum completes 10 cycles in $8 \, \text{seconds}$, so the period is: $T = \frac{8}{10} = 0.8 \, \text{seconds}$ The amplitude is half of the total distance, so: $A = \frac{40}{2} = 20 \, \text{cm}$

(b) Describe the transformations to transform $y = \sin x$.

The transformations are:

1. Vertical scaling: The amplitude is $20$, so multiply by $20$.
2. Horizontal scaling: The period is $T = 0.8$, so $B = \frac{2\pi}{T} = \frac{2\pi}{0.8} = \frac{5\pi}{2}$.
3. No vertical or horizontal shifts are mentioned.

The transformed equation is: $y = 20 \sin \left( \frac{5\pi}{2} t \right)$

(c) Write the equation of motion.

Using the above, the equation for the pendulum’s motion is: $y(t) = 20 \sin \left( \frac{5\pi}{2} t \right)$

Would you like me to sketch the graphs or explain any part further? Here are some additional questions to deepen your understanding:

1. How do you calculate the amplitude of a periodic function like this?
2. What physical aspects of the pendulum correspond to the amplitude and period?
3. How does the vertical shift in Problem 9 affect the blood pressure graph?
4. How would the range change if the amplitude of the blood pressure increased to $30$?
5. Why is cosine used instead of sine for the blood pressure model?

Tip: Remember, for sinusoidal functions, the vertical shift represents the midpoint of oscillation!