Problem 79 - Algebra and Trigonometry with Analytic Geometry

Question

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 $$ R=2.07 \ln x-2.04 \text { provided } 0 \leq R \leq 1 . $$ For example, if \(R=0.65\), then there is a \(65 \%\) chance that a woman will have a heart attack over an average lifetime. Calculate \(R\) for a female with \(C=242\) and \(H=78\).

Step-by-Step Solution

Answer

The lifetime risk \(R\) is approximately 0.303.

Step 1: Identify Known Values

We know the total amount of cholesterol, \(C = 242\), and the amount of high-density lipoprotein cholesterol, \(H = 78\). We are to find the ratio \(x = \frac{C}{H}\).

Step 2: Calculate Ratio x

Calculate \(x\) using the formula \(x = \frac{C}{H}\). Substitute the known values: \[ x = \frac{242}{78} \].

Step 3: Compute the Ratio x

Perform the division to find \(x\): \[ x = \frac{242}{78} \approx 3.1026 \].

Step 4: Substitute x into the Risk Formula

Insert the calculated ratio \(x\) into the formula for lifetime risk \(R\). This gives us: \[ R = 2.07 \ln(3.1026) - 2.04 \].

Step 5: Calculate the Natural Logarithm

Compute \(\ln(3.1026)\) using a calculator, which gives approximately 1.1321.

Step 6: Calculate Risk R

Substitute the value of the logarithm obtained in Step 5 into the equation: \[ R = 2.07 \times 1.1321 - 2.04 \].

Step 7: Solve for R

Calculate \(R\) by performing the multiplication and then the subtraction: \[ R = 2.34265 - 2.04 = 0.30265 \].

Step 8: Verify that R is within Bounds

We must ensure \(0 \leq R \leq 1\). Here, \(R \approx 0.30265\), which satisfies this condition.

Key Concepts

Cholesterol Calculation

Calculating cholesterol is an essential part of assessing the risk of heart disease. The ratio of total cholesterol (C) to high-density lipoprotein cholesterol (H), known as HDL, is key. This ratio is denoted by x and is calculated using the formula:

• \( x = \frac{C}{H} \)

An example can help make this clearer. Suppose the total cholesterol (C) is 242, and HDL cholesterol (H) is 78.
We calculate the ratio as:

• \( x = \frac{242}{78} \)

After performing the division, the cholesterol ratio is approximately 3.1026.
Remember, a healthier cholesterol ratio should ideally be lower, reducing the risk of heart disease as high ratios are often linked to increased risk.

Natural Logarithm

The natural logarithm, often simply called the ln, is a mathematical function that helps to transform complex calculations into simpler ones.It is denoted as \( \ln(x) \) and is the inverse function of exponentiation.
Why is it useful in calculating lifetime risk for heart disease? The natural logarithm helps to moderate how changes in cholesterol ratios translate into changes in risk. This is because heart disease progression doesn't increase linearly with cholesterol—it accelerates at higher levels.
To calculate the ln of our cholesterol ratio x\(\) = \(3.1026\):

• \( \ln(3.1026) \approx 1.1321 \)

Using this value in further calculations helps to establish a more accurate risk assessment.

Lifetime Risk Formula

To estimate the lifetime risk of a heart attack for females based on cholesterol ratios, we use a specialized formula:

• \( R = 2.07 \ln x - 2.04 \)

This formula calculates R, which indicates the lifetime risk as a percentage once multiplied by 100.
Let's calculate R given the previously found ln(3.1026):
Substitute into the formula:

• \( R = 2.07 \times 1.1321 - 2.04 \)
• \( R = 2.34265 - 2.04 \)
• \( R \approx 0.30265 \)

This result means there's an approximate 30.27% lifetime risk. Ensuring R falls between 0 and 1 confirms the calculation is valid. This range is crucial for real-world applicability, ensuring meaningful risk values.