Cholesterol level in women Studies relating serum...

Cholesterol level in women Studies relating serum cholesterol level to coronary heart disease suggest that a risk factor is the ratio $x$ of the total amount $C$ of cholesterol in the blood to the amount $H$ of high-density lipoprotein cholesterol in the blood. For a female, the lifetime risk $R$ of having a heart attack can be approximated by the formula $$ R=2.07 \ln x-2.04 \quad \text { provided } \quad 0 \leq R \leq 1 $$ For example, if $R=0.65,$ then there is a $65 \%$ chance that a woman will have a heart attack over an average lifetime. (a) Calculate $R$ for a female with $C=242$ and $H=78$ (b) Graphically estimate $x$ when the risk is $75 \% .$

Cholesterol level in women Studies relating serum cholesterol level to coronary heart disease suggest that a risk factor is the ratio $x$ of the total amount $C$ of cholesterol in the blood to the amount $H$ of high-density lipoprotein cholesterol in the blood. For a female, the lifetime risk $R$ of having a heart attack can be approximated by the formula
$$
R=2.07 \ln x-2.04 \quad \text { provided } \quad 0 \leq R \leq 1
$$
For example, if $R=0.65,$ then there is a $65 \%$ chance that a woman will have a heart attack over an average lifetime.
(a) Calculate $R$ for a female with $C=242$ and $H=78$
(b) Graphically estimate $x$ when the risk is $75 \% .$

Key Concepts

Cholesterol Ratio as a Risk Factor

In risk assessment, ratios such as the total cholesterol to HDL cholesterol ratio are used to standardize measurements and provide a relative measure of risk. This approach helps in interpreting individual cholesterol levels, as it reflects both the harmful and beneficial components of blood lipids, thereby serving as a significant indicator for the likelihood of coronary heart disease.

Logarithmic Functions in Health Risk Modeling

Logarithmic functions, like the natural logarithm used in the model, play a key role in transforming non-linear relationships into linear ones. In the context of health risk modeling, the logarithm helps relate an increase in a risk factor (such as the cholesterol ratio) to the increase in the probability of an adverse event, making the relationship easier to analyze and interpret.

Graphical Estimation Techniques

Graphical estimation is a method used to approximate the values of variables in an equation by visually inspecting the plotted relationship between variables. This technique is especially useful in instances where an exact algebraic solution may not be straightforward, as it allows for the estimation of input values corresponding to specific outcomes based on the curve of the function.

Domain and Range in Risk Functions

When modeling probabilities with functions, it is essential to consider the valid input (domain) and output (range) values. For risk functions, ensuring that the output is constrained between 0 and 1 ensures that the predicted probabilities are meaningful and accurately reflect the likelihood of an event occurring, maintaining consistency with real-world interpretations of risk.

Transcript

00:01 In this problem of logarithmic function, we have given that study is relating serum cholesterol level to the coronary heart disease suggests that a risk factor is the ratio x.

00:13 So this is the risk factor ratio x of the total amount c of the cholesterol in the blood, of the high amount h of the high density lycoprotein cholesterol in the blood.

00:28 This is here x equals to c divided with h.

00:32 For a female, the lifetime risk r of having a heart attack can be approximated by the formula.

00:41 So this is r is equal to this is the lifetime risk.

00:44 R is represented by lifetime risk.

00:48 So r is given by 0.

00:50 This is 2 .07 ln x minus 2 .04.

00:58 So this is the risk problem and this is modeled by the formula that is equal to 2 .704.

01:04 07 lns minus 2 .04 provided that is r is from 0 to 1.

01:12 For example r is equal to 065.

01:16 That means if r is 0 .65 that means the risk is 65 % chance that a woman will have a heart attack over an average lifetime.

01:27 So this would be 65 % if r is equal to 0 .65 that mean this is 65 % chance that women will have a heart attack and now we have to calculate we have to calculate r for a female for which r is here c is equal to 242 and h is equal to 78 and now we have given that x is equal to c divided with h so this would be x is equals to 242 divided with 78 so this is when we calculate this is 242 divided with 78 so this is when we calculate this is 242 divided with 708 so this is 78 is equal to 3 .102.

02:10 So this is here 3 .102.

02:13 That means we have to solve for x is equal to 3 .102.

02:16 So put the value...

Please give Ace some feedback