Problem 54 - Calculus

Question

A controversial 1992 Danish study \(^{9}\) reported that men's average sperm count has decreased from 113 million per milliliter in 1940 to 66 million per milliliter in \(1990 .\) (a) Express the average sperm count, \(S\), as a linear function of the number of years, \(t,\) since \(1940 .\) (b) A man's fertility is affected if his sperm count drops below about 20 million per milliliter. If the linear model found in part (a) is accurate, in what year will the average male sperm count fall below this level?

Step-by-Step Solution

Answer

(a) \(S = -0.94t + 113\), (b) The sperm count drops below 20 million/ml in 2039.

Step 1: Identify Variables and Initial Conditions

Given the initial sperm count in 1940 is 113 million/ml and in 1990 it is 66 million/ml, we need to express these in terms of the variable \(S\) for sperm count and \(t\) for the number of years since 1940. Therefore, for 1940: \(S = 113\) and \(t = 0\), and for 1990: \(S = 66\) and \(t = 50\).

Step 2: Calculate the Slope of the Linear Function

The sperm count decreases linearly. We use the two data points to find the slope (rate of change) \(m\). The formula for the slope \(m\) is:\[ m = \frac{S_2 - S_1}{t_2 - t_1} = \frac{66 - 113}{50 - 0} = \frac{-47}{50} = -0.94 \]Thus, the sperm count decreases by 0.94 million/ml per year.

Step 3: Create the Linear Equation

Using the slope-point form of a line, \(y = mx + b\), use the point \((0, 113)\):\[ S = -0.94t + 113 \]This represents the average sperm count as a function of years since 1940.

Step 4: Set Up Equation to Find the Year When S = 20

We need to find \(t\) when \(S < 20\). Substitute \(S = 20\):\[ 20 = -0.94t + 113 \]Rearrange to solve for \(t\):\[ 0.94t = 113 - 20 \]\[ 0.94t = 93 \]

Step 5: Solve for t

Divide both sides by 0.94:\[ t = \frac{93}{0.94} \approx 98.94 \]Round to the nearest whole number, so \(t = 99\).

Step 6: Determine the Year

Since \(t = 99\) represents the number of years since 1940, the corresponding year is:\[ 1940 + 99 = 2039 \]

Key Concepts

Slope Calculation

The slope of a line is a measure of how steep a line is, indicating the rate of change between any two points on the line.
It is calculated by finding the change in the y-values (vertical change) over the change in the x-values (horizontal change).

• The formula for slope, often represented by the letter \(m\), is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Think of \( y_2 \) and \( y_1 \) as two different outputs (or results) of the function, and \( x_2 \) and \( x_1 \) as the two different inputs (or years, in the exercise).In our example, these are sperm counts from 1940 and 1990. Let's break this down:

• We started with a sperm count of 113 million/ml in 1940 (\( t = 0 \))
• And it decreased to 66 million/ml by 1990 (\( t = 50 \)). Using the point values provided, we substitute them into the formula to calculate the slope:
• \[m = \frac{66 - 113}{50 - 0} = -0.94\]

This negative slope means the sperm count is decreasing by 0.94 million/ml each year.

Point-Slope Form

The Point-Slope Form of a line is an equation that helps you write the equation of a line when you know one point on the line and the slope.
It is expressed as:

• \[y - y_1 = m(x - x_1)\]

In this example, we use this form to create the equation of the average sperm count.

• We already found the slope \( m = -0.94 \) (the rate of decrease in sperm count per year).
• We use the point from 1940, which is \((x_1, y_1) = (0, 113) \), where \( y_1 \) is the sperm count. We plug these values into the point-slope formula:
• \[S - 113 = -0.94(t - 0)\]

By simplifying, we convert this into the slope-intercept form:\[S = -0.94t + 113\].
This represents the average sperm count \(S\) over time \(t\) since 1940.

Equation of a Line

An equation of a line in a two-dimensional space gives you a complete description of that line in terms of its slope and the y-intercept.
The y-intercept is where the line crosses the y-axis, and in the slope-intercept form, is represented by \(b\).For our linear function expressed in the form

• \[y = mx + b\]

• The slope \(m\) tells us the rate of change of the sperm count. In this case, it's \(-0.94\).
  It tells us that each year, the sperm count reduces by 0.94 million/ml.
• The y-intercept \(b\) here is 113, which is the starting average sperm count in 1940.
  This is where the function would intersect the \(y\)-axis when \(t = 0\).

The equation for this scenario, \(S = -0.94t + 113\), provides a detailed snapshot of the average sperm count trend over the years.
It's useful for predicting future values of sperm count by simply plugging in different \(t\) values (number of years since 1940).
For example, if you want to find out when the sperm count drops below 20 million/ml, solve for \(t\) when \(S = 20\):

• \[ 20 = -0.94t + 113 \]
• This rearranges to: \[0.94t = 93\] Divide by 0.94 to find \(t\).
• \[t \approx 98.94 \]

Rounding gives 99 years since 1940, which would make it 2039. This method shows how you can use a linear function to make predictions using algebraic manipulation.