Problem 32 - Algebra 2

Question

For Exercises 32 and \(33,\) use the following information. The average height (in inches) for boys ages 1 to 20 can be modeled by the equation \(B(x)=-0.001 x^{4}+0.04 x^{3}-0.56 x^{2}+5.5 x+25\) , where \(x\) is the age (in years). The average height for girls ages 1 to 20 is modeled by the equation \(G(x)=-0.0002 x^{4}+0.006 x^{3}-0.14 x^{2}+3.7 x+26\) . Graph both equations by making a table of values. Use \(x=\\{0,2,4,6,8,10,\) \(12,14,16,18,20 \\}\) as the domain. Round values to the nearest inch.

Step-by-Step Solution

Answer

Plot boys' and girls' height values on a graph from the calculated values for each given age.

Step 1: Choose the Domain Values

Identify the domain values given in the exercise for both equations, which are: \(x = \{ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \}\). These values represent the age in years.

Step 2: Calculate Values for Boys' Height

Use the equations for boys' height, \(B(x) = -0.001x^4 + 0.04x^3 - 0.56x^2 + 5.5x + 25\), and substitute each domain value to find the corresponding height.Calculate and round each result to the nearest inch:- \(B(0) = 25\)- \(B(2) = 35\)- \(B(4) = 42\)- \(B(6) = 48\)- \(B(8) = 53\)- \(B(10) = 57\)- \(B(12) = 62\)- \(B(14) = 66\)- \(B(16) = 68\)- \(B(18) = 70\)- \(B(20) = 71\)

Step 3: Calculate Values for Girls' Height

Use the equations for girls' height, \(G(x) = -0.0002x^4 + 0.006x^3 - 0.14x^2 + 3.7x + 26\), and substitute each domain value to find the corresponding height.Calculate and round each result to the nearest inch:- \(G(0) = 26\)- \(G(2) = 33\)- \(G(4) = 39\)- \(G(6) = 45\)- \(G(8) = 50\)- \(G(10) = 55\)- \(G(12) = 60\)- \(G(14) = 64\)- \(G(16) = 67\)- \(G(18) = 69\)- \(G(20) = 70\)

Step 4: Draw Graph Using Calculated Values

On graph paper, plot the corresponding heights for each \(x\) value as points for boys and girls. Label the \(x\)-axis as 'Age in Years' and \(y\)-axis as 'Height in Inches'. Use different colors or markers for boys' and girls' data points, and connect the points to illustrate the average growth trends for both.

Key Concepts

Graphing

Graphing is a vital tool in visualizing mathematical models, like those describing average height. In this exercise, graphing helps us see how the average height of boys and girls varies with age. To graph these polynomial functions:

• Set up a coordinate grid with the x-axis representing age in years and the y-axis representing height in inches.
• Plot each pair of (age, height) coordinates calculated from Boys’ and Girls’ models.
• By plotting these points for ages 0 through 20, we visualize growth trends.

Connecting these points gives a curve that visually demonstrates the changing height over time. This graphical representation allows us to easily compare the growth patterns of boys and girls side-by-side.

Modeling

Modeling with polynomial functions provides an effective method to predict real-world phenomena, like average height based on age. The given equations for boys and girls are fourth degree polynomials, which capture complex variations in growth patterns. The coefficients of each polynomial term

• Determine the curve's shape.
• Reflect real-life impacting factors.

For example, the term with x⁴ represents slower changes over the life span, while higher degree terms help reflect accelerated growth phases. Using these models, we learn how height trends shift gradually over twenty years, a span covering significant growth stages in children. Understanding modeling helps us see how predictions and data points coalesce into meaningful insights.

Average height

Average height denotes the mean height for a particular age group. It helps summarize how tall boys and girls generally are during various stages of childhood and adolescence. Using the mathematical models provided:

• Boys & Girls height equations reveal average heights if given an age input.
• Values are rounded to the nearest inch for simplicity.

With data points recorded for each age, these average heights allow us to gauge developmental milestones, compare growth lengths, and interpret shifts in heights that occur over these formative years. This average serves as a key indicator of physical development.

Domain values

Domain values are the set of permissible inputs for a function. In this exercise, they are specific ages from 0 to 20 years. Choosing these domain values is essential to adequately modeling functions. For the height functions:

• Domain directly relates to practical, real-world constraints (ages where growth occurs).
• This range represents developmental years where these height functions are applicable.

By focusing on these specific domain values, we ensure that the polynomial models accurately reflect the intended age group. Understanding domain values informs us about the function’s relevant application, ensuring that our analyses remain within appropriate bounds.