Problem 74 - 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 192 grams?

Step-by-Step Solution

Answer

The fetus would weigh 192 grams after 8 weeks since \(t = \sqrt{64} = 8\).

Step 1 - Setting equation to given weight

To start off solving, one needs to set the given weight model \(W=3 t^{2}\) equal to the weight we want to find the time for, which is 192 grams. So the equation becomes \(192 = 3 t^{2}\).

Step 2 - Rearrange the equation

Rearrange the equation for easy computation, get \(t^{2}\) alone on one side of the equation. We do this by dividing through the equation by 3 to get \(t^{2} = 64\).

Step 3 - Solve for time

Solve for the time \(t\). As \(t^{2} = 64\), taking square root of both sides will give us the value of \(t\) (the time). Hence, \(t = \sqrt{64}\).

Key Concepts

What are Weight Equations?

In algebraic modeling, weight equations are used to express the relationship between weight and other variables, such as time. In the given exercise, the formula \(W=3t^2\) is a weight equation, where \(W\) represents the weight of a human fetus in grams, and \(t\) signifies the time in weeks. Weight equations help in tracking and predicting changes in weight over time by providing a mathematical model.

These equations are valuable in various fields such as biology, physics, and medicine. They translate complex real-world phenomena into simpler, analyzable forms using mathematical language.

• Weight equations typically involve variables and constants.
• They can be rearranged to solve for unknown variables.
• They provide insights into the growth patterns of living organisms.

Understanding weight equations involves recognizing how changes in one variable, like time, can directly affect another, such as weight. This understanding enables us to make informed predictions and decisions.

Understanding Square Roots

Square roots are fundamental to solving quadratic equations like the one in the exercise. If you have an equation in the form of \(t^2 = 64\), the square root helps us find possible values for \(t\). The square root of 64 is 8, which means \(t = 8\).

But what exactly does taking the square root mean? The square root of a number is a value that, when multiplied by itself, gives the original number. In our case, \(8 \times 8 = 64\). Here’s a simple breakdown:

• Square roots help solve for the original variable in second-degree equations.
• Every positive number actually has two square roots: one positive and one negative (e.g., \(\pm 8\)), although negative time doesn't apply here.
• Using square roots is a key step in solving quadratic equations.

Mastering square roots is crucial to dealing with quadratic expressions and equations smartly and effectively.

Rearranging Equations for Problem Solving

Rearranging equations is like solving a puzzle. It's about getting the variable you want to solve for by itself on one side of the equation. In our exercise, we needed to rearrange the equation \(192 = 3t^2\) to solve for \(t\).

Here's how you go about it: Divide both sides by 3 to isolate \(t^2\). This gives \(t^2 = 64\). This step effectively simplifies the problem, making it easier to solve.

• Rearranging involves inverse operations to isolate the variable.
• Simplifies complex equations to more manageable forms.
• Crucial for understanding and solving algebraic expressions.

This process is essential in algebraic modeling and numerous other mathematical problems. It enhances your ability to think critically and solve a wide array of mathematical challenges.