Problem 88 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

According to the Jenss model for predicting the height of preschool children, the rate of growth of a typical preschool child is $$ R(t)=25.8931 e^{-0.993 t}+6.39 \quad\left(\frac{1}{4} \leq t \leq 6\right) $$ centimeters/year, where \(t\) is measured in years. The height of a typical 3 -mo-old preschool child is \(60.2952 \mathrm{~cm}\). a. Find a model for predicting the height of a typical preschool child at age \(t\). b. Use the result of part (a) to estimate the height of a typical 1-yr-old child.

Step-by-Step Solution

Answer

The model for predicting the height of a typical preschool child at age t is \(H(t) = 26.0866 e^{-0.993 t} + 6.39t + 41.6913\). Using this model, the estimated height of a typical 1-year-old child is approximately 67.8565 cm.

Step 1: Convert 3 months to years

Since t is measured in years, we first need to convert the 3-month-old child's age into years. 3 months = 0.25 (1 year = 12 months) So the 3-month-old child's age in years is 0.25.

Step 2: Integrate the rate function to find the height function

We are given the rate function R(t), and we need to find the height function H(t). Since the rate is nothing but the derivative of height with respect to time, we need to integrate R(t) with respect to t. Let H(t) be the height function. Then, \(H'(t) = R(t) = 25.8931 e^{-0.993 t} + 6.39\) Now integrate both sides with respect to t: \(H(t) = \int (25.8931 e^{-0.993 t} + 6.39) dt\)

Step 3: Calculate the integration

Calculate the integral: \(H(t) = -25.8931 e^{-0.993 t} \frac{1}{-0.993} + 6.39t + C\) Rewrite the height function: \(H(t) = 26.0866 e^{-0.993 t} + 6.39t + C\)

Step 4: Find the constant of integration using the given height

We know that the height of a 3-month-old child is 60.2952 cm. So when t=0.25, H(t)=60.2952. Plug these values into the height function to get the constant C: 60.2952 = 26.0866 e^{-0.993*0.25} + 6.39 * 0.25 + C Then solve for C: C = 60.2952 - 26.0866 e^{-0.993*0.25} - 6.39 * 0.25 C ≈ 41.6913

Step 5: Write the final height function

Now we have the constant C, and we can write the height function as: \(H(t) = 26.0866 e^{-0.993 t} + 6.39t + 41.6913\) This is the model for predicting the height of a typical preschool child at age t. #b. Use the result of part (a) to estimate the height of a typical 1-year-old child#

Step 6: Plug in t=1

To find the height of a 1-year-old child, plug in t=1 into the height function from part (a): \(H(1) = 26.0866 e^{-0.993*1} + 6.39*1 + 41.6913\)

Step 7: Calculate the height

Calculate the height for t=1: H(1) ≈ 67.8565 cm The height of a typical 1-year-old child is approximately 67.8565 cm.

Key Concepts

Mathematical Modeling

Mathematical modeling is a powerful tool for understanding the world around us. By creating a mathematical representation of a real-world phenomenon, we can analyze, predict, and even control various aspects of the situation. In the exercise we've examined, the Jenss model uses mathematical equations to represent the growth rate of preschool children.

An effective model captures the essence of what it represents and can make accurate predictions. The growth rate model in our exercise, represented by the function R(t), is crafted from observed data and biological understanding of human growth. It provides us a way to predict the future height of a child, given their current age and initial height. Taking into account that growth can be complex and vary between individuals, mathematical models must be adaptable and subject to refinement as new data becomes available.

Exponential Functions

Exponential functions play a crucial role in modeling situations where growth or decay is proportional to the current value - such as in population dynamics or radioactive decay. In our example, the rate of growth function R(t) involves an exponential decay term, which suggests that as children grow older, their growth rate decreases in a specific way.

These functions are written as \( a \times e^{kx} \), where \( e \) is the base of the natural logarithm, \( a \) is a constant that represents the starting value, and \( k \) is the growth (or decay) rate. The negative sign in the exponent of the exponential term in R(t) indicates a decay over time, a common trait in physical and biological processes as they approach a limit or equilibrium.

Integrals in Applied Mathematics

Integrals are essential in applied mathematics and are used to aggregate quantities over a period. In the context of growth prediction in children, integration allows us to move from the rate of growth (a rate per time) to total growth (a cumulative measure).

When we integrate the growth rate function over time, we find the height function H(t), reflecting total height at any given age. The constant of integration, C, aligns the general solution of the integral to the specific scenario, incorporating initial conditions like the known height of the child at a certain age. This makes the model personalized and far more practical for actual predictions and reflects an important principle in applied mathematics: integrals often require context-specific information to provide meaningful results.