Cholesterol Researchers have found that the risk of coronary...

Cholesterol Researchers have found that the risk of coronary heart disease rises as blood cholesterol increases. This risk may be approximated by the function $$R(c)=3.19(1.006)^{c}, \quad 100 \leq c \leq 300$$ where $R$ is the risk in terms of coronary heart disease incidence per 1000 per year, and $c$ is the cholesterol in mg/dL. Suppose a person's cholesterol is 180 $\mathrm{mg} / \mathrm{d} \mathrm{L}$ and going up at a rate of 15 $\mathrm{mg} / \mathrm{dL}$ per year. At what rate is the person's risk of coronary heart disease going up? Source: Circulation.

Cholesterol Researchers have found that the risk of coronary heart disease rises as blood cholesterol increases. This risk may be approximated by the function
$$R(c)=3.19(1.006)^{c}, \quad 100 \leq c \leq 300$$
where $R$ is the risk in terms of coronary heart disease incidence per 1000 per year, and $c$ is the cholesterol in mg/dL. Suppose a person's cholesterol is 180 $\mathrm{mg} / \mathrm{d} \mathrm{L}$ and going up at a rate of 15 $\mathrm{mg} / \mathrm{dL}$ per year. At what rate is the person's risk of coronary heart disease going up? Source: Circulation.

Key Concepts

Related Rates

Related rates problems involve determining the rate at which one quantity changes by relating it to another quantity whose rate of change is known. This concept is commonly applied in situations where multiple variables are interdependent and their rates of change with respect to time are connected.

Chain Rule

The chain rule is a fundamental differentiation technique used for calculating the derivative of composite functions. It states that the derivative of a function composed of another function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function, making it essential for solving problems where one variable depends on another.

Exponential Functions

Exponential functions are characterized by a variable exponent and are used to model processes that exhibit constant percentage growth or decay. Their derivatives involve the original exponential function multiplied by the natural logarithm of the base, which is critical when assessing the change rate in processes described by exponential models.

Transcript

00:01 Okay, so this is question 54 from section 4 .4.

00:03 One of the biggest things here before we start solving the equation and it's understanding what we're asking here.

00:11 So we know that we have this model, which is the risk for coronary heart disease, and we know it rises as blood cholesterol increases.

00:23 R is the risk, and c is the cholesterol.

00:26 And we want to suppose a person's cholesterol, is 180.

00:32 So this is just for the purpose of the question.

00:35 We want to assume it's 180.

00:37 And we can't plug that in yet because when we plug that in, we would have what the risk is.

00:43 But we also know it's going up at a rate of 15.

00:47 So we know that the rate of change, the rate of change dc for the cholesterol is equal to 15.

01:04 So we have c and we have the rate that the c is changing and we want to know what rate the risk is going up so the rate is the derivative so we can represent that as dr d t or just as r prime so our prime is equal to dr d t so and we're trying to find the rate so we want to know the rate when we have these conditions for c and dct so we know that to find a rate a rate of change essentially we need to find the derivative of r of c first so to find this derivative we're going to have 3 .19 and then this right here is in the form of a number raised to some g and if we think back to the beginning of this chapter if we have an equation in this form we need to take the derivative it's the natural log of the number times the function times the derivative of whatever that g is.

02:13 In this case our g is c.

02:15 So you can replace the g here with c.

02:18 So we have 3 .19 and then we need the derivative of the number race to the c which is times the natural log of the number times the whole function times the derivative of g which is the derivative of c but we don't really know what c is so we just write that as an arbitrary derivative for right now.

02:45 So this is what we have and now we have to solve this equation we have c and we have the rate...

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