Problem 67 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to Exercise 66. The researchers at the Institute also measured the thickness (also in millimeters) of the cortex of children of age \(t\) yr who were of average intelligence. These data lead to the model $$ \begin{array}{r} A(t)=-0.00005 t^{3}-0.000826 t^{2}+0.0153 t+4.55 \\ (5 \leq t \leq 19) \end{array} $$ Show that the cortex of children with average intelligence reaches maximum thickness at age \(6 \mathrm{yr}\).

Step-by-Step Solution

Answer

To find the maximum thickness of the cortex, we first find the derivative of the function \(A(t) = -0.00005t^3 - 0.000826t^2 + 0.0153t + 4.55\) with respect to t, which is \(A'(t) = -0.00015t^2 - 0.001652t + 0.0153\). Setting the derivative equal to 0, we use the quadratic formula to get two possible values of t: \(t \approx 6.022\) and \(t \approx 16.985\). Since the given age range is 5 ≤ t ≤ 19, we can conclude that the cortex of children with average intelligence reaches maximum thickness at age 6 years (t ≈ 6.022), which lies in the given range.

Step 1: Find the derivative of A(t) with respect to t

To find the maximum thickness of the cortex, we first need to find the first derivative of the function A(t). This is given by: \(A(t) = -0.00005t^3 - 0.000826t^2 + 0.0153t + 4.55\) Taking the derivative with respect to t: \(A'(t) = \frac{d}{dt} (-0.00005t^3 - 0.000826t^2 + 0.0153t + 4.55)\) Now we can apply the respective power rules of the derivatives: \(A'(t) = -0.00015t^2 - 0.001652t + 0.0153\)

Step 2: Set the derivative equal to 0 and solve for t

To find the maximum thickness, we want to find the value of t for which the derivative becomes 0. So, let's set A'(t) = 0: \(-0.00015t^2 - 0.001652t + 0.0153 = 0\) Now, we can solve for t. Since the equation is a quadratic equation, we can use the quadratic formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Here, a = -0.00015, b = -0.001652, and c = 0.0153 \(t = \frac{0.001652 \pm \sqrt{(-0.001652)^2 - 4(-0.00015)(0.0153)}}{2(-0.00015)}\) Calculating the values, we get two possible values of t: \(t \approx 6.022, t \approx 16.985\)

Step 3: Verify that the maximum thickness is attained at the specified age range

We have been given an age range of 5 ≤ t ≤ 19. Since t ≈ 6.022 lies within that range, we can conclude that the cortex of children with average intelligence reaches maximum thickness at age 6 years. The value t ≈ 16.985 will give the local minimum point in this case, which is not what we are looking for.

Key Concepts

Derivative of a Function

When attempting to find the maximum thickness of a cortex, one needs to understand the derivative of a function. A derivative represents the rate at which a function's output changes as its input changes. In simpler terms, it's like examining the speedometer of a car; the derivative tells you how fast the car's speed is changing at any given moment.

In applied mathematics, particularly when dealing with physical measurements like thickness over time, derivatives play a crucial role. They allow us to determine when growth occurs, levels off, or declines—in this case, when a child's cortex is increasing in thickness and when it reaches its maximum before thinning out.

The process begins by taking the original function, which is a model representing the thickness of the cortex relative to the child's age, and finding its derivative with respect to time. This derivative function can then inform us about the behavior of the cortex thickness over time, by showing where its rate of change is zero, which indicates potential maximum or minimum points.

Quadratic Formula

In the given exercise, after finding the derivative of the cortex thickness model, the resulting equation resembles a quadratic function. A quadratic function is a second-degree polynomial, typically written in the form of ax^2 + bx + c. To determine the specific time at which thickness reaches its maximum, solving the quadratic equation is necessary.

The quadratic formula is a staple in algebra and comes in handy in this scenario. This formula, written as \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), provides the solution(s) for t when the derivative of the function is set to zero. The formula represents two possible solutions, where the plus-minus symbol (\(\pm\)) denotes that you can either add or subtract the square root term.

When it comes to real-world applications like monitoring the development of a child's brain, the quadratic formula is invaluable because it gives exact points in time at which important changes occur. By plugging in the coefficients from our derivative into the quadratic formula, we can find the ages at which the thickness of the cortex either peaks or has a local minimum.

Applied Mathematics

Applied mathematics is the branch of mathematics that deals with mathematical methods used in practical applications. In the context of the cortex thickness problem, applied mathematics helps form a mathematical model—like the one given by function A(t)—to represent real-world phenomena.

The model used for predicting cortex thickness is derived from data analysis and is subject to verification and validation through mathematical techniques. Once the model is established, methods from calculus, such as finding derivatives and solving quadratic equations, are used to answer questions about specific properties, in this case, the age when the cortex thickness is at its maximum.

Applied mathematicians often use derivatives to optimize real-life systems, such as determining the most efficient size for a package or the best time to perform an action to maximize efficiency or minimize costs. Here, they helped find the crucial age of cortex development, which can inform further scientific research or medical assessments.

Power Rule in Derivatives

The power rule is a basic yet powerful tool used to differentiate functions in calculus. It states that the derivative of \( t^n \) with respect to t is \( n \cdot t^{n-1} \). This is how we derive the function \( A(t) \) presented in the exercise. Each term of the function is taken separately, and the power rule is applied.

Let's revisit the first step of our solution:

• The term \( -0.00005t^3 \), applying the power rule, becomes \( -0.00015t^2 \).
• Similarly, the term \( -0.000826t^2 \) becomes \( -0.001652t \).
• And the linear term \( 0.0153t \), when differentiated, loses its exponent and remains a constant—\( 0.0153 \).

Understanding and applying the power rule helps simplify the derivative process significantly and is fundamental when dealing with polynomial functions. It allows quick computation of derivatives, which are essential for finding maxima and minima in various applications, from engineering to economics.