According to the Jenss model for predicting the height of...

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.

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.

Key Concepts

Mathematical Modeling

Mathematical modeling involves formulating a mathematical representation of a real-world situation in order to predict or understand its behavior. In the context of growth measurements, this process constructs functional relationships that describe changes in height over time based on observed growth rates.

Differential Equations

Differential equations are equations that relate a function with its derivative, serving as essential tools in modeling processes that change continuously, such as growth. Here, the given rate function, which is the derivative of the height function, sets up a differential equation for height with respect to time.

Integration and Antiderivatives

Integration is the method of finding the antiderivative of a function, which reverses differentiation. By integrating the given growth rate function, one obtains the height function expressed in terms of an integral plus a constant, illustrating how cumulative growth can be deduced from instantaneous growth rates.

Initial Conditions

Initial conditions are specific known values at the start of the process that are used to determine any constants that appear during integration of a differential equation. In growth modeling, knowing the initial height at a certain time allows the unique determination of the height function that accurately represents the situation.

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