Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

The growth of a fetus more than 12 weeks old can be approximated by the formula \(L=1.53 t-6.7\), where \(L\) is the length (in centimeters) and \(t\) is the age (in weeks). Prenatal length can be determined by ultrasound. Approximate the age of a fetus whose length is 28 centimeters.

Step-by-Step Solution

Answer

The approximate age of the fetus is 22.68 weeks.

Step 1: Write Down the Given Formula

We need to use the formula given in the problem, which is \(L = 1.53t - 6.7\), where \(L\) is the length in centimeters and \(t\) is the age in weeks.

Step 2: Substitute Given Length

We know the length \(L\) is 28 centimeters. Substitute 28 for \(L\) in the formula: \(28 = 1.53t - 6.7\).

Step 3: Solve for t

To find the age \(t\), we need to solve the equation for \(t\). First, add 6.7 to both sides of the equation: \(28 + 6.7 = 1.53t\). This simplifies to \(34.7 = 1.53t\).

Step 4: Isolate t

Divide both sides of the equation by 1.53 to isolate \(t\): \(t = \frac{34.7}{1.53}\).

Step 5: Calculate the Age

Perform the division to calculate \(t\): \(t \approx 22.68\).

Key Concepts

Linear Equations

Linear equations are essential tools in mathematics that help establish relationships between quantities. In the context of fetal growth calculations, linear equations express how variables interact in a predictable manner. A linear equation, like the one used for calculating fetal growth, takes the general form: \(y = mx + b\). Here, \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope, and \(b\) is the y-intercept. In our exercise, the equation \(L = 1.53t - 6.7\) shows us how to model fetal length as a dependent outcome based on the independent variable, age. The slope (1.53) represents the rate of change in length per week, while the y-intercept (-6.7) theoretically would be the length at zero weeks, providing context within the model. By understanding how to manipulate linear equations, we can predict or find unknown values effectively.Steps to solve a linear equation:

• Identify the equation and what you need to find (e.g., \(t\) when \(L\) is given).
• Substitute the known values into the equation.
• Solve for the unknown variable through algebraic manipulation.
• Check your results for consistency with the context of the problem.

Understanding and using linear equations help in many real-life scenarios beyond prenatal studies, like financial calculations and physics predictions.

Prenatal Development

Prenatal development covers the intricate process by which a fetus develops from conception to birth. It encompasses several stages, with fetal growth being a pivotal focus during the second and third trimesters. The fetal stage, from 9 weeks to birth, sees rapid growth and development. By 12 weeks, critical structures and systems have formed, marking the start of the stage where growth is more easily measured and tracked, often through mathematical models like the one provided in our exercise. Key aspects of prenatal development:

• Cell Division and Specialization: An embryo's cells multiply and begin to take specific roles.
• Organ Formation: Major organs start forming in the embryonic stage and mature during the fetal phase.
• Growth: Continuous size increase, as depicted by the linear equation used for fetal length.

Understanding these stages allows healthcare providers to monitor fetal health and development effectively.

Ultrasound Measurements

Ultrasound measurements provide non-invasive ways to monitor fetal growth and development. This technology uses sound waves to create images of the fetus inside the womb. Those images are then used to measure aspects like fetal length, head circumference, and more. Why use ultrasound measurements?

• Safe and Non-Invasive: Ultrasound is a safe method with no exposure to ionizing radiation.
• Accuracy: Provides crucial information about the fetus's development, which can be used alongside mathematical models to estimate factors like age or due date.
• Monitoring Growth: Helps in tracking whether the fetus is growing at a typical rate compared to standards.

For our exercise, the length obtained from ultrasound serves as a part of the equation to estimate fetal age, underscoring the invaluable role that ultrasound technology plays in prenatal care.