Question

1 - andersons school is 8 miles directly south of his house. his work is directly west of his home. the path from home to school forms a 60° angle with the path from work to school. what is the distance from school to work? 13.86 miles 4 miles the distance cannot be determined with the given information 16 miles

Explanation:

Step1: Identify the triangle type

Since Anderson's school is south of his house and work is west of his house, the triangle formed by home - school - work is a right - triangle. Let \(H\) be home, \(S\) be school, \(W\) be work. \(\angle HSW = 60^{\circ}\), \(HS = 8\) miles. We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\) in a right - triangle. Here, if we consider \(\theta = 60^{\circ}\) and the adjacent side to \(\theta\) is \(HS\) and the hypotenuse is \(WS\) (wait, no, actually, using trigonometric ratios for right - triangle. Let's use \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). If we consider the right - triangle \(HSW\) with \(\angle HSW = 60^{\circ}\) and \(HS\) as the adjacent side to \(\angle HSW\) and \(SW\) as the hypotenuse. Wait, no, better to use \(\cos60^{\circ}=\frac{HS}{SW}\). But actually, we know that in a right - triangle, if we assume the right - angle at \(H\) (home), \(\cos60^{\circ}=\frac{HS}{SW}\).

Step2: Apply the cosine formula

We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Given \(\theta = 60^{\circ}\), \(\cos60^{\circ}=\frac{1}{2}\), and the adjacent side (from school to home) \(HS = 8\) miles. Let the distance from school to work be \(x\). Using \(\cos60^{\circ}=\frac{HS}{x}\) (where \(x\) is the distance from school to work). Substituting \(\cos60^{\circ}=\frac{1}{2}\) and \(HS = 8\) into \(\cos60^{\circ}=\frac{HS}{x}\), we get \(\frac{1}{2}=\frac{8}{x}\). Cross - multiplying gives \(x = 16\) miles.

Another way: Using the right - triangle relationship. If we consider the right - triangle with one side \(HS = 8\) (south - home to school), and the angle between school - home and school - work is \(60^{\circ}\). If we use the formula \(SW=\frac{HS}{\cos60^{\circ}}\) (since \(\cos\alpha=\frac{\text{adjacent}}{\text{hypotenuse}}\), here \(\alpha = 60^{\circ}\), adjacent is \(HS\), hypotenuse is \(SW\)).

Answer:

16 miles