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January 2018, Part IV

This Question in Part IV is worth 6 credits. All work must be shown or explained for full credit. A correct numerical answer without work is only worth 1 credit.


37. The resting blood pressure of an adult patient can be modeled by the function P below, where P(t) is the pressure in millimeters of mercury after time t in seconds.

P(t) = 24cos(3πt) + 120
On the set of axes below, graph y = P(t) over the domain 0 ≤ t ≤ 2.

Determine the period of P. Explain what this value represents in the given context.

Normal resting blood pressure for an adult is 120 over 80. This means that the blood pressure oscillates between a maximum of 120 and a minimum of 80. Adults with high blood pressure (above 140 over 90) and adults with low blood pressure (below 90 over 60) may be at risk for health disorders. Classify the given patient’s blood pressure as low, normal, or high and explain your reasoning.

Answer:
The 3π means that the period is going to be 2π/3π, or 2/3. Note that there are six boxes between 0 and 1.
The 24 and 120 tell you that the maximum is going to be at 120 + 24 = 144 and the minimum is going to be 120 - 24 = 96.
Cosine tells that at 0, P(0) = 144, as will 2/3, 4/3, and 6/3 (or 2).
The minimums will occur at 1/3, 3/3 (or 1), and 5/3.
The function will equal 120 at each sixth in between the max and min: 1/6, 3/6, 5/6, 7/6, 9/6, and 11/6.

See the image below:

The patient's blood pressure is 144/96, which is more than 140/90, so he would be classified as having high blood pressure.



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