Researchers have found that the risk of coronary heart disease rises as blood cholesterol increases. This risk may be approximated by the function ,
, where is the risk in terms of coronary heart disease incidence per 1000 per year, and c is the cholesterol in . Suppose a person’s cholesterol is 176
and going up at a rate of per year. At what rate is the person’s risk of coronary heart disease going up?
The person’s risk of coronary heart disease is going up at the rate of coronary heart disease incidence per 1000 per year per mg/dL of cholesterol.
(Round to three decimal places as needed.)

Ask by Lambert Campos. in the United States

Feb 14,2025

Question

Researchers have found that the risk of coronary heart disease rises as blood cholesterol increases. This risk may be approximated by the function ,
, where is the risk in terms of coronary heart disease incidence per 1000 per year, and c is the cholesterol in . Suppose a person’s cholesterol is 176
and going up at a rate of per year. At what rate is the person’s risk of coronary heart disease going up?
The person’s risk of coronary heart disease is going up at the rate of coronary heart disease incidence per 1000 per year per mg/dL of cholesterol.
(Round to three decimal places as needed.)

Ask by Lambert Campos. in the United States

Feb 14,2025

UpStudy AI · Accepted Answer

We are given the risk function

R(c) = 3.169·(1.006)^c,

with cholesterol c in mg/dL (100 ≤ c ≤ 300) and risk measured in coronary heart disease (CHD) incidence per 1000 per year. If the cholesterol level is increasing over time, then c is a function of time and we must use the chain rule to compute the rate at which the risk R changes with respect to time.

Step 1. Compute the derivative of R with respect to c.

dR/dc = d/dc [3.169·(1.006)^c]  
     = 3.169·(1.006)^c·ln(1.006).

Step 2. Use the chain rule to compute dR/dt.

dR/dt = (dR/dc)·(dc/dt).

We are told that at the moment considered,
  c = 176 mg/dL  and  dc/dt = 15 mg/dL per year.

Thus,

dR/dt = 3.169·(1.006)^(176)·ln(1.006) · 15.

Step 3. Compute the numerical value.

(a) Find ln(1.006).  
  ln(1.006) ≈ 0.005976.

(b) Compute (1.006)^(176).  
  (1.006)^(176) = exp(176·ln(1.006)) ≈ exp(176·0.005976) ≈ exp(1.051) ≈ 2.860.

(c) Now put these into the derivative:

dR/dt ≈ 3.169 · 2.860 · 0.005976 · 15.

Let's compute step by step:

(i) 3.169 · 2.860 ≈ 9.055.  
(ii) 9.055 · 0.005976 ≈ 0.05407.  
(iii) 0.05407 · 15 ≈ 0.811.

Step 4. Interpret the result.

The computed derivative dR/dt ≈ 0.811, which means that at the moment when the cholesterol is 176 mg/dL and increasing at a rate of 15 mg/dL per year, the risk of coronary heart disease is increasing at a rate of about 0.811 incidence per 1000 people per year.

Thus, the answer (rounded to three decimal places) is:

0.811 coronary heart disease incidence per 1000 per year.

This is the rate at which the person's risk is going up.
The person's risk of coronary heart disease is increasing at a rate of 0.811 incidence per 1000 per year.

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