Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Growth of children The Jenss model is generally regarded as the most accurate formula for predicting the height of preschool children. If \(y\) is height (in centimeters) and \(x\) is age (in years), then $$ y=79.041+6.39 x-e^{3.261-0.993 x} $$ for \(\frac{1}{4} \leq x \leq 6\). From calculus, the rate of growth \(R\) (in \(\mathrm{cm} /\) year) is given by \(R=6.39+0.993 e^{3261-0.993 x}\). Find the height and rate of growth of a typical 1-year-old child.

Step-by-Step Solution

Answer

The height is approximately 75.768 cm and the rate of growth is about 15.981 cm/year at age 1.

Step 1: Identify the variables

We need to find the height of a typical 1-year-old child (\(y\) for \(x = 1\)) and the rate of growth at the same age (\(R\) for \(x = 1\)). Here, \(x = 1\) represents the age.

Step 2: Calculate the height for x = 1

Using the Jenss model formula \(y = 79.041 + 6.39x - e^{3.261 - 0.993x}\), substitute \(x = 1\):\[y = 79.041 + 6.39(1) - e^{3.261 - 0.993(1)}\]Calculate each term separately:- First term: \(79.041\)- Second term: \(6.39 \times 1 = 6.39\)- Third term: \(e^{3.261 - 0.993} = e^{2.268}\)Using a calculator, \(e^{2.268} \approx 9.663\).Now compute the height:\[y = 79.041 + 6.39 - 9.663 \approx 75.768\]So, the height \(y\) is approximately 75.768 centimeters.

Step 3: Calculate the rate of growth for x = 1

The rate of growth formula is \(R = 6.39 + 0.993 e^{3.261 - 0.993x}\). Substitute \(x = 1\):\[R = 6.39 + 0.993 e^{3.261 - 0.993(1)}\]We already calculated \(e^{3.261 - 0.993} = 9.663\) in Step 2.Calculate each term:- First term: \(6.39\)- Second term: \(0.993 \times 9.663 \approx 9.591\)Now compute the rate of growth:\[R = 6.39 + 9.591 \approx 15.981\]So, the rate of growth \(R\) is approximately 15.981 cm/year.

Key Concepts

Preschool Growth Prediction

Predicting the growth of preschool children can be a challenging task, but the Jenss model provides a reliable way to estimate it. This model is specifically designed to capture the growth patterns of children during their early years. Its importance lies in its ability to account for the rapid changes in height that occur during this developmental stage.
The Jenss model is expressed as:

• \[ y = 79.041 + 6.39x - e^{3.261 - 0.993x} \]

Here, \(y\) stands for the height in centimeters, and \(x\) represents age in years. By substituting a specific value of age into the equation, we can predict a child's expected height. This is critical for pediatricians and parents who are interested in monitoring a child's growth trajectory.
Understanding how to use this formula allows for effective tracking of height changes and can highlight potential growth issues if deviations from the predicted pattern are observed.

Calculus Rate of Change

The concept of the rate of change is intertwined with calculus and plays a crucial role in understanding dynamic systems, including the growth of children. In the context of the Jenss model, the rate of change reflects how quickly a child's height increases over time.
The formula for the rate of growth is given by:

• \[ R = 6.39 + 0.993 e^{3.261 - 0.993x} \]

The equation from calculus allows us to understand not only how tall a child is at a specific age but also how quickly they are growing at that age. For example, at age 1, the calculated rate of growth is approximately 15.981 cm/year. This means that during the first year, the child's height is increasing at an average rate of roughly 16 centimeters per year. Knowing such rates helps in assessing whether a child is growing too slowly or too quickly and if any interventions might be needed.
The use of calculus here simplifies complex growth behavior into a smooth function that we can easily analyze.

Exponential Function

Exponential functions often describe processes where changes occur at a rate proportional to the current state, such as in natural growth processes. In the Jenss model, the exponential term \( e^{3.261 - 0.993x} \) is crucial in modeling how children's growth decelerates over time.
In this exponential term:

• \(e\) represents Euler's number, approximately 2.718, which is a constant base for natural logarithms.
• The power \((3.261 - 0.993x)\) indicates how the growth's rapidity changes with age.

At younger ages, \(x\) is small, and the exponential term contributes larger values, indicating fast growth. As \(x\) increases, the value of the exponential function diminishes, reflecting the slower growth rate typically observed as children age.
The predictive power of exponential functions in growth models like the Jenss model helps visualize and anticipate the naturally decreasing growth rate and ensures that height predictions remain realistic and grounded in biological processes.