2. La Ley de Hooke establece que, dentro de ciertos límites, hay una relación lineal entre la fuerza mathrm(F) ejercida por un resorte y la longitud l de éste. Es decir, que existen números reales alpha y beta tales que F=a l+beta . Usar los siguientes datos para estimar las constantes alpha y beta : Longitud (mathrm(cm)) & Fuerza (mathrm(N)) 8.89 & 4.5 10.2 & 9.8 11.4 & 12.5 12.7 & 19.1

## Step 1: Understand the Problem<br />### We are given a linear relationship between force \( F \) and length \( l \) of a spring, expressed as \( F = \alpha l + \beta \). Our task is to estimate the constants \( \alpha \) and \( \beta \) using the provided data points.<br /><br />## Step 2: Set Up the System of Equations<br />### Using the given data points, we can set up a system of equations. Each pair of (length, force) gives us an equation:<br />- For \( l = 8.89 \), \( F = 4.5 \): \( 4.5 = \alpha \cdot 8.89 + \beta \)<br />- For \( l = 10.2 \), \( F = 9.8 \): \( 9.8 = \alpha \cdot 10.2 + \beta \)<br />- For \( l = 11.4 \), \( F = 12.5 \): \( 12.5 = \alpha \cdot 11.4 + \beta \)<br />- For \( l = 12.7 \), \( F = 19.1 \): \( 19.1 = \alpha \cdot 12.7 + \beta \)<br /><br />## Step 3: Solve the System of Equations<br />### To find the best estimates for \( \alpha \) and \( \beta \), we can use the method of least squares, which minimizes the sum of the squares of the residuals (the differences between the observed and predicted values).<br /><br />### The normal equations for least squares in this context are:<br />\[<br />\begin{align*}<br />\sum l_i^2 \cdot \alpha + \sum l_i \cdot \beta &= \sum l_i \cdot F_i \\<br />\sum l_i \cdot \alpha + n \cdot \beta &= \sum F_i<br />\end{align*}<br />\]<br />where \( n \) is the number of data points.<br /><br />### Calculating these sums from the data:<br />- \( \sum l_i = 8.89 + 10.2 + 11.4 + 12.7 = 43.19 \)<br />- \( \sum F_i = 4.5 + 9.8 + 12.5 + 19.1 = 45.9 \)<br />- \( \sum l_i^2 = 8.89^2 + 10.2^2 + 11.4^2 + 12.7^2 = 1006.0466 \)<br />- \( \sum l_i \cdot F_i = 8.89 \cdot 4.5 + 10.2 \cdot 9.8 + 11.4 \cdot 12.5 + 12.7 \cdot 19.1 = 514.63 \)<br /><br />### Substituting these into the normal equations:<br />\[<br />\begin{align*}<br />1006.0466 \cdot \alpha + 43.19 \cdot \beta &= 514.63 \\<br />43.19 \cdot \alpha + 4 \cdot \beta &= 45.9<br />\end{align*}<br />\]<br /><br />## Step 4: Solve the Linear System<br />### Solving these equations simultaneously (using methods such as substitution or matrix operations), we find:<br />- \( \alpha \approx 2.05 \)<br />- \( \beta \approx -13.73 \)