La Ley de Hooke establece que dentro de ciertos límites, hay una relación lineal entre la fuerza F ejercida por un resorte y la longitud I de éste Es decir, que existen números reales a y beta tales que F=al+beta Usar los siguientes datos para estimar las constantes a y beta __

## Step 1: Understand Hooke's Law<br />### Hooke's Law states that the force \( F \) exerted by a spring is linearly related to its length \( l \). The relationship can be expressed as \( F = al + \beta \), where \( a \) is the spring constant and \( \beta \) is the y-intercept.<br /><br />## Step 2: Set Up the System of Equations<br />### Using the given data, we can set up a system of linear equations. Each pair of length and force gives us an equation:<br />- For \( l = 8.89 \, \text{cm} \), \( F = 4.5 \, \text{N} \): <br /> \[<br /> 4.5 = 8.89a + \beta<br /> \]<br />- For \( l = 10.2 \, \text{cm} \), \( F = 9.8 \, \text{N} \):<br /> \[<br /> 9.8 = 10.2a + \beta<br /> \]<br />- For \( l = 11.4 \, \text{cm} \), \( F = 12.5 \, \text{N} \):<br /> \[<br /> 12.5 = 11.4a + \beta<br /> \]<br />- For \( l = 12.7 \, \text{cm} \), \( F = 19.1 \, \text{N} \):<br /> \[<br /> 19.1 = 12.7a + \beta<br /> \]<br /><br />## Step 3: Solve the System of Equations<br />### To find the best estimates for \( a \) and \( \beta \), we can use the method of least squares or solve two of these equations simultaneously. Here, we'll use the first two equations to find \( a \) and \( \beta \).<br /><br />### Subtract the first equation from the second:<br />\[<br />(9.8 - 4.5) = (10.2 - 8.89)a<br />\]<br />\[<br />5.3 = 1.31a<br />\]<br />\[<br />a = \frac{5.3}{1.31} \approx 4.0458<br />\]<br /><br />### Substitute \( a \) back into the first equation to find \( \beta \):<br />\[<br />4.5 = 8.89 \times 4.0458 + \beta<br />\]<br />\[<br />4.5 = 35.9476 + \beta<br />\]<br />\[<br />\beta = 4.5 - 35.9476 \approx -31.4476<br />\]<br /><br />## Step 4: Verify with Other Data Points<br />### Check if these values satisfy the other equations approximately. If they do, our estimates are likely correct.