Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

A baby weighs 10 pounds at birth, and three years later the child's weight is 30 pounds. Assume that childhood weight \(W\) (in pounds) is linearly related to age \(t\) (in years). (a) Express \(W\) in terms of \(t\). (b) What is \(W\) on the child's sixth birthday? (c) At what age will the child weigh 70 pounds? (d) Sketch, on a \(t W\)-plane, a graph that shows the relationship between \(W\) and \(t\) for \(0 \leq t \leq 12\).

Step-by-Step Solution

Answer

(a) \( W = \frac{20}{3}t + 10 \). (b) 50 pounds. (c) 9 years old.

Step 1: Understand the linear relationship

The problem states that the child's weight is linearly related to age, which implies a straight-line equation of the form \( W = mt + b \), where \( m \) is the slope and \( b \) is the y-intercept (initial weight).

Step 2: Establish two points for the line

We have two data points: at birth (\( t = 0 \)), the weight \( W = 10 \), and at age three (\( t = 3 \)), the weight \( W = 30 \). These points can be written as \( (0, 10) \) and \( (3, 30) \).

Step 3: Calculate the slope (m)

The formula for the slope between two points \( (t_1, W_1) \) and \( (t_2, W_2) \) is \( m = \frac{W_2 - W_1}{t_2 - t_1} \). Substituting, we get \( m = \frac{30 - 10}{3 - 0} = \frac{20}{3} \).

Step 4: Write the equation of the line

Using the slope \( m = \frac{20}{3} \) and the y-intercept \( b = 10 \) from the point \( (0, 10) \), the equation of the line is \( W = \frac{20}{3}t + 10 \). This answers part (a).

Step 5: Calculate weight at age 6

Substitute \( t = 6 \) into the equation \( W = \frac{20}{3}t + 10 \): \[ W = \frac{20}{3} \times 6 + 10 = 40 + 10 = 50. \] The child's weight at age 6 is 50 pounds, which answers part (b).

Step 6: Determine age for weight of 70

Set \( W = 70 \) in the equation \( W = \frac{20}{3}t + 10 \) and solve for \( t \): \[ 70 = \frac{20}{3}t + 10. \] Solve for \( t \): \[ 70 - 10 = \frac{20}{3}t, \] \[ 60 = \frac{20}{3}t, \] \[ t = \frac{60 \times 3}{20} = 9. \] The child will weigh 70 pounds at age 9, which answers part (c).

Step 7: Sketch the graph

The graph is a straight line starting from \( W = 10 \) at \( t = 0 \) to \( W = 70 \) at \( t = 9 \). Extend the graph linearly and label axes: x-axis as age (t) and y-axis as weight (W). The line passes through points (0, 10), (3, 30), and (9, 70). Ensure the graph extends to at least \( t = 12 \).

Key Concepts

Slope and Intercept

When you're dealing with linear equations, understanding the slope and intercept is crucial. The slope, often represented as "m," is a measure of how much one variable changes in relation to another. In our scenario, it tells us how much the child's weight increases each year. To compute the slope from two points on a line, use the formula: \( m = \frac{W_2 - W_1}{t_2 - t_1} \). This gives you the rate of change over time.
The intercept, represented as "b," is the starting value of the dependent variable when the independent variable is zero. In the context of this problem, the intercept is the baby's birth weight, 10 pounds, when age \( t = 0 \). Thus, the equation of the line, representing the weight over time, will have the form \( W = mt + b \). It helps you quickly determine the weight given any age by plugging \( t \) into the equation.

Linear Functions

Linear functions describe relationships where one variable changes at a constant rate with respect to another. They are visually represented by straight lines on a graph.
Linear functions are defined by equations of the form \( W = mt + b \), where each term has a specific role:

• Slope (m): This indicates how steep the line is and whether it rises or falls as you move along the x-axis (here, age \( t \)).
• Intercept (b): This is where the line meets the y-axis, setting the initial state or starting point of the observation (here, birth weight).

The beauty of linear functions is their predictability and simplicity; once you know the slope and intercept, you can easily predict future values or assess past values.

Problem Solving

Problem solving with linear functions involves using given data to create and utilize an equation for predictions and analysis.

• Form an Equation: From the given points, \((0, 10)\) and \((3, 30)\), the slope was derived as \( \frac{20}{3} \) and the intercept as 10. This results in \( W = \frac{20}{3}t + 10 \).
• Make Predictions: To find the weight at age 6, plug 6 into the equation: \( W = \frac{20}{3} \times 6 + 10 \), which calculates to 50 pounds.
• Backtrack Ages: To find the age at which the child reaches a certain weight, solve \( 70 = \frac{20}{3}t + 10 \). This method provides the age of 9 years when the weight is 70 pounds.

These calculations are useful for estimating growth trends or identifying specific growth milestones using a linear model.

Graph Interpretation

Interpreting graphs of linear functions is critical to visualizing relationships and changes. A linear graph shows how weight varies with age.

• Set Axes: Mark the x-axis as age and the y-axis as weight. This setup highlights the relationship and changes over time.
• Plot Points: For example, points such as \((0, 10)\), \((3, 30)\), and \((9, 70)\) should be marked. These points are evidence of the linear relationship.
• Draw the Line: A straight line through these points represents the continuous increase in weight over time.
• Analyze Slope: The slope of this line shows that for every additional year, the expected weight increases by \( \frac{20}{3} \) pounds.

By sketching and analyzing the line, you can gain insights into trends and predict future events, such as a child's growth milestones.