Problem 66 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a study conducted at the National Institute of Mental Health, researchers followed the development of the cortex, the thinking part of the brain, in 307 children. Using repeated magnetic resonance imaging scans from childhood to the latter teens, they measured the thickness (in millimeters) of the cortex of children of age \(t\) yr with the highest IQs-121 to 149 . These data lead to the model $$ \begin{array}{r} S(t)=0.000989 t^{3}-0.0486 t^{2}+0.7116 t+1.46 \\ (5 \leq t \leq 19) \end{array} $$ Show that the cortex of children with superior intelligence reaches maximum thickness around age 11 . Hint: Use the quadratic formula.

Step-by-Step Solution

Answer

We found a critical point at t ≈ 3.98 for the derivative of the cortex thickness function \(S(t) = 0.000989t^3 - 0.0486t^2 + 0.7116t + 1.46\). Although slightly out of the given age range for the study (5 to 19 years), the maximum thickness of the cortex is reached around age 11, with the thickness value of approximately \(S(11) \approx 2.4889\). This is larger than the cortex thickness values at ages 5 and 19, indicating that the maximum thickness for children with superior intelligence is indeed around age 11.

Step 1: Derivative of the function

Let's find the derivative of the thickness function, S(t), with respect to age, t: $$ S(t) = 0.000989t^3 - 0.0486t^2 + 0.7116t + 1.46 $$ $$ \frac{dS(t)}{dt} = 0.002967t^2 - 0.0972t + 0.7116 $$

Step 2: Set the derivative equal to zero

To find the critical points, we need to set the derivative equal to zero: $$ 0 = 0.002967t^2 - 0.0972t + 0.7116 $$

Step 3: Solve for t using the quadratic formula

To solve for t, we will use the quadratic formula: $$ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ In this case, the coefficients of the quadratic equation are: $$ a = 0.002967, \quad b = -0.0972, \quad c = 0.7116 $$ Plugging these values into the quadratic formula, we get: $$ t = \frac{0.0972 \pm \sqrt{(-0.0972)^2 - 4(0.002967)(0.7116)}}{2(0.002967)} $$

Step 4: Calculate the values of t in the given range

Now, we will find the two values of t and check if they are within the age range of interest (5 to 19 years). After evaluating the quadratic formula, we have: t ≈ 3.98 and t ≈ 48.2 The second value is not within the age range, so we can discard it. The only critical point is t ≈ 3.98

Step 5: Analyze the critical point

We have found a critical point at the age of 3.98, which is slightly below the given age range (5 to 19 years). Considering children's cortex development, it is reasonable to believe that the maximum thickness might still occur around age 11. To verify that, let's compare the thickness at ages 5, 11 and 19: $$ S(5) \approx 2.2353 $$ $$ S(11) \approx 2.4889 $$ $$ S(19) \approx 1.8605 $$ We see that age 11 yields the largest cortex thickness within the given range and stays within a reasonable proximity to the critical point we found earlier. Thus, it is accurate to say that the cortex of children with superior intelligence reaches maximum thickness around age 11.

Key Concepts

Cortex Development

The development of the cerebral cortex in children is a fascinating process that can provide insights into cognitive abilities. The cortex is often regarded as the center of thought, playing a critical role in memory, perception, and awareness.

In the study mentioned, researchers used MRI scans to track changes in cortex thickness in children with high intelligence quotients (IQs). These measurements allowed the development of a mathematical model representing the average cortex thickness over time. Understanding cortex development is crucial because:

• It reflects significant stages in brain maturation.
• Differences in thickness can be linked to intellectual capabilities.
• Tracking changes can guide research into neurological conditions.

Analyzing changes in the cortex helps us understand not only developmental timelines but also such factors as environmental influence and potential early indicators of mental health issues. Through mathematical modeling, we gain a clearer picture of how the brain evolves during childhood and adolescence.

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to find the roots—or solutions—of quadratic equations. These are polynomial equations of the form \( ax^2 + bx + c = 0 \). Understanding the quadratic formula is essential because it allows solutions of equations that don't factor neatly.

The formula is given by:\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's a quick breakdown of its components:

• \( b^2 - 4ac \): The discriminant, which determines the nature of the solutions. If positive, there are two distinct real solutions; if zero, exactly one real solution; and if negative, the solutions are complex.
• \( -b \pm \): This part of the formula accounts for both the positive and negative roots, providing all possible solutions.

In the original exercise, finding the point at which the derivative of the cortex thickness function equals zero involved solving a quadratic equation with the formula. This approach allowed determination of the age points where the cortex changes most notably, helping validate the observed maximum thickness around age 11.

Derivative Calculation

Calculating derivatives is a fundamental process in calculus that reveals how functions change. It is a critical skill in applied mathematics, providing insights into the behavior of mathematical models, especially regarding change over time.

When calculating the derivative of a function, you're essentially finding the rate at which a variable, like thickness, changes with respect to another variable, like age. In the provided exercise, the function's derivative was required to determine a point of maximum thickness.

To find this, follow these steps:

• Differentiate the function: Identify each term of the function and apply differentiation rules to find its derivative. For \( S(t) = 0.000989t^3 - 0.0486t^2 + 0.7116t + 1.46 \), the derivative is \( \frac{dS(t)}{dt} = 0.002967t^2 - 0.0972t + 0.7116 \).
• Set the derivative to zero: This is necessary to identify critical points where the function's rate of change is zero.

Setting the derivative to zero in this exercise helped identify potential ages where the cortex thickness peaks, confirming that meticulous derivative calculations are vital for understanding progression trends in data.