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We are given the function for blood pressure as:

\[
P(t) = -20 \cos(300^\circ t) + 100
\]

To find the rate of change of blood pressure at \( t = 3 \) seconds, we need to calculate the derivative of \( P(t) \), which gives us the rate of change of the blood pressure with respect to time. Then, we'll evaluate the derivative at \( t = 3 \).

The steps to solve this are:
1. Differentiate \( P(t) \) with respect to \( t \).
2. Evaluate the derivative at \( t = 3 \).

I'll start by computing the derivative.The derivative of the blood pressure function \( P(t) \) with respect to time \( t \) is:

\[
P'(t) = \frac{1080000 \sin\left(\frac{54000 t}{\pi}\right)}{\pi}
\]

Now, evaluating the derivative at \( t = 3 \) seconds gives:

\[
P'(3) = \frac{1080000 \sin\left(\frac{162000}{\pi}\right)}{\pi}
\]

This is the exact form of the rate of change of blood pressure at \( t = 3 \) seconds. I can simplify it further to find a numerical value. Let me calculate that for you.The rate of change of blood pressure at \( t = 3 \) seconds is approximately \( 34,233.23 \, \text{mm Hg/s} \).

Would you like further details or clarifications on any part of this solution?

Here are five related questions to explore:
1. How does the cosine function affect the rate of change of blood pressure over time?
2. What is the physical significance of the rate of change being positive or negative in this context?
3. How would the period of the function affect the time when the maximum or minimum blood pressure occurs?
4. What happens to the blood pressure as time increases indefinitely in this model?
5. How can this model be used to estimate average blood pressure over a cycle?

**Tip:** When differentiating trigonometric functions, always remember to apply the chain rule carefully, especially when the argument of the function involves a constant multiple of the variable.

For a person at rest, the function P(t) = -20 cos(300*t) + 100 models blood pressure, in millimeters of mercury (mm Hg), at time t seconds. What is the rate of change in blood pressure at t = 3 s?

Learn how to find the rate of change of blood pressure modeled by the function P(t) = -20 cos(300*t) + 100. This calculus problem involves the derivative of a trigonometric function, applying the chain rule, and evaluating it at t = 3 seconds. Suitable for students in Grade 12 or college-level calculus.

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