Question

mathematics—session 2: calculator

1. the table shows the height of a child, in centimeters, at different ages.

height of a child

age (years)	height (cm)
3	90.5
5	105.0
7	119.0
8	125.5
12	151.5

create an equation to model the relationship between the height, in centimeters, and the age, in years, of this child. show your work or explain your equation. be sure to include the limitations of your equation.
enter your equation, your work or explanation, and the limitations in the box provided.

Explanation:

Step1: Check linearity (calculate slopes)

First, we calculate the slope between different points. Let's take two points, say \((2, 84.5)\) and \((3, 90.5)\). The slope \(m\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\). So, \(m=\frac{90.5 - 84.5}{3 - 2}=\frac{6}{1} = 6\). Let's check another pair, \((5, 105.0)\) and \((7, 119.0)\). \(m=\frac{119.0 - 105.0}{7 - 5}=\frac{14}{2}=7\). Wait, maybe a better approach is to use linear regression or assume a linear model (even though slopes vary a bit, we can find a best - fit line). Let's use the two - point formula for a line \(y - y_1=m(x - x_1)\). Let's take the first point \((2, 84.5)\) and the last point \((12, 151.5)\). The slope \(m=\frac{151.5 - 84.5}{12 - 2}=\frac{67}{10}=6.7\).

Step2: Use point - slope form

Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(2, 84.5)\) and \(m = 6.7\). We have \(y-84.5=6.7(x - 2)\).

Step3: Simplify the equation

Expand the right - hand side: \(y-84.5 = 6.7x-13.4\). Then, add \(84.5\) to both sides: \(y=6.7x - 13.4 + 84.5\), so \(y=6.7x + 71.1\).

Step4: Check with other points

Let's check for \(x = 3\): \(y=6.7\times3+71.1=20.1 + 71.1 = 91.2\), which is close to \(90.5\). For \(x = 5\): \(y=6.7\times5+71.1=33.5+71.1 = 104.6\), close to \(105.0\). For \(x = 7\): \(y=6.7\times7+71.1=46.9+71.1 = 118\), close to \(119.0\). For \(x = 8\): \(y=6.7\times8+71.1=53.6+71.1 = 124.7\), close to \(125.5\). For \(x = 12\): \(y=6.7\times12+71.1=80.4+71.1 = 151.5\), which matches.

Step5: State the limitations

The equation \(y = 6.7x+71.1\) (where \(y\) is height in cm and \(x\) is age in years) is a linear model. Limitations: The growth of a child is not strictly linear over a long period. For example, during puberty, the growth rate changes. Also, the model is based on a limited set of data points (only 6 points), and it may not accurately predict height for ages outside the range \(2\leq x\leq12\) (ages less than 2 or greater than 12 may have different growth patterns).

Answer:

Equation: \(y = 6.7x+71.1\) (where \(y\) represents height in centimeters and \(x\) represents age in years)
Work/Explanation: We calculated the slope between the points \((2, 84.5)\) and \((12, 151.5)\) as \(m=\frac{151.5 - 84.5}{12 - 2}=6.7\). Then we used the point - slope form \(y - y_1=m(x - x_1)\) with the point \((2, 84.5)\) to get \(y-84.5 = 6.7(x - 2)\), and simplified it to \(y = 6.7x+71.1\). We verified the equation with other data points and found it to be a good fit for the given data.
Limitations: The model is linear, but a child's growth is not strictly linear over all ages (e.g., growth rate changes during puberty). Also, the model is based on a small set of data points and may not be accurate for ages outside the range \(2\leq x\leq12\) (ages less than 2 or greater than 12).