Problem 73 - Introductory Algebra for College Students

Question

The weight of a human fetus is modeled by the formula \(W=3 t^{2}\) where \(W\) is the weight, in grams, and \(t\) is the time, in weeks, with \(0 \leq t \leq 39 .\) Use this formula to solve Exercises \(73-74\) After how many weeks does the fetus weigh 108 grams?

Step-by-Step Solution

Answer

The fetus weighs 108 grams after 6 weeks.

Step 1: Identify the Known Variables

The weight \(W\) of the fetus is given as 108 grams.

Step 2: Substitute the Known Variable into the Formula

Substitute \(W = 108\) into the given formula \(W=3t^{2}\), to get \(108=3t^{2}\).

Step 3: Solve the Equation for 't'

First, divide both sides of the equation by 3 to isolate \(t^{2}\) on one side, i.e. \(36 = t^{2}\). Then, take the square root of both sides to solve for 't'. Note that we consider only the positive square root as time cannot be negative. Making this operation we obtain \(t = 6\).

Key Concepts

Algebraic Modeling

Algebraic modeling is a powerful tool for representing real-world situations through mathematical equations and expressions. It provides a simplified version of a scenario that can be analyzed and understood using algebraic techniques. In our example, the relationship between the growth of a fetus and time is represented by an algebraic model: the formula W=3t^2. This model helps us to predict the fetal weight based on the time of growth in weeks, presenting a clear, quantifiable relationship between these two variables.

To construct an effective model, it is crucial to identify the variables at play and their interconnections. In fetal weight growth modeling, we're fundamentally interested in two variables: weight and time. These are denoted by W for weight and t for time, respectively. By observing patterns or using scientific knowledge, we can determine a functional form, like W=3t^2, which signifies that weight is a function of the square of time multiplied by some constant. This constant, 3 in this case, is derived from empirical data or fits established by biological research on fetal growth.

Quadratic Equations

Quadratic equations are mathematical expressions of the form ax^2 + bx + c = 0, where x represents an unknown variable and a, b, and c are constants with a ≠ 0. These types of equations are second-degree because the highest power of the variable is two (x^2).

The scenario presented in the exercise is an example of a quadratic equation in disguise. The given formula, W=3t^2, becomes a quadratic equation when we set W to a specific value such as 108 grams. The equation 108=3t^2 or 3t^2 - 108 = 0 is then solved by isolating the term with the variable t. The solutions of a quadratic equation are found using various methods: factoring, completing the square, or applying the quadratic formula. In the given problem, the solution involves simple manipulation—dividing by 3 and then taking the square root of both sides—which is sufficient due to the straightforward nature of the equation.

Concept of Variables

In algebra, variables are symbols that represent unknown or variable quantities. They are placeholders for values that can change or that we're attempting to solve for. Understanding the concept of variables is fundamental in grasping any algebraic expression or equation.

In the context of our fetal weight growth model, W stands for the weight of the fetus in grams, and t represents the time in weeks. These variables allow for the algebraic formula W=3t^2 to be versatile—it can be used to find the weight at various points in time or to determine the time at which a certain weight is achieved. When we're solving for t, given that W=108 grams, t is the unknown we're trying to find, and it is isolated through algebraic manipulation.

Recognizing the roles variables play in an equation helps differentiate between dependent and independent variables. In our equation, W is the dependent variable since its value depends on the value of t, the independent variable. As such, the time measured in weeks will determine the weight, reflecting a direct dependency which is fundamental in understanding how changes in one variable affect another within mathematical models.