Problem 23 - College Algebra

Question

For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year. When did the baby weight 10.4 pounds?

Step-by-Step Solution

Answer

The baby weighed 10.4 pounds approximately between the 5th and 6th months, reaching more than 10.4 pounds in month 6.

Step 1: Understand the Initial Condition

The problem states that the weight of the newborn is initially 7.5 pounds at birth, which we'll consider as the starting point.

Step 2: Calculate the Monthly Weight Gain

The baby gains 0.5 pounds every month. This means each month, the baby will weigh 0.5 pounds more than the previous month.

Step 3: Set Up the Equation for the Weight Goal

Let the number of months after birth be represented by the variable \( x \). The weight \( W \) of the baby after \( x \) months can be modeled as follows: \[ W = 7.5 + 0.5x \] The target weight is 10.4 pounds. Thus, we have the equation: \[ 7.5 + 0.5x = 10.4 \]

Step 4: Solve the Equation

Subtract 7.5 from both sides of the equation: \[ 0.5x = 10.4 - 7.5 \]This simplifies to:\[ 0.5x = 2.9 \] Now, divide both sides by 0.5 to solve for \( x \): \[ x = \frac{2.9}{0.5} = 5.8 \]

Step 5: Interpret the Solution

Since \( x \) represents months, a value of 5.8 doesn't align perfectly with whole months, which are needed in real life. Therefore, we round up to the next whole month. Hence, the baby would weigh more than 10.4 pounds at 6 months.

Key Concepts

Monthly Weight Gain

Understanding how a baby's weight changes monthly allows us to predict future growth patterns. In this scenario, the newborn started at a specific weight and then consistently gained weight every month. Here, the baby gains 0.5 pounds a month.
This steady weight increase can be expressed as a linear progression, where each month adds an incremental amount to the previous total. This concept of linear growth is significant in the formation of equations used to model weight gain over time, providing a calculation model for various periods such as weeks or months.

Initial Conditions

At the start, every mathematical problem needs a clear starting point or initial condition. In our scenario, the weight of the newborn is 7.5 pounds. This starting condition sets a baseline from which all changes are measured.
Initial conditions are vital because they establish the context for any problem. Without a defined starting point, predictions and calculations would lack accuracy and relevance. Recognizing these initial conditions helps students understand how and why certain calculations are necessary throughout the problem-solving process.

Solving Equations

Solving equations involves finding the value of unknown variables that make the equation true. In our baby’s weight gain problem, we set up an equation to determine when the baby would weigh a specific amount, 10.4 pounds. The equation used is:

• \( W = 7.5 + 0.5x \)
• This models the baby's weight with respect to time in months.
• To find \( x \), where the weight \( W \) is 10.4, we adjust the equation: \( 7.5 + 0.5x = 10.4 \).

The goal is to isolate \( x \). By solving, we find \( x \) equals 5.8, meaning it takes approximately 5.8 months for the baby to reach 10.4 pounds. This involves basic algebra, using subtraction and division to solve for the unknown.

Interpreting Solutions

Interpreting solutions means understanding what the solution tells us in real-life terms. Although we solved for \( x \) and found it to be 5.8, we must consider the practical applicability. In real-world scenarios, round numbers make sense—months, days, etc., are whole numbers.
Since you can't have a fraction of a month when considering a baby's growth stages, we round up to 6 months to ensure the baby reaches more than the goal weight of 10.4 pounds. Interpreting solutions not only confirms that the mathematical aspects are correct but ensures that they make sense within the real-life context they are applied to.