Question

jake and maddie each take the closest path from their homes to the store. based on the picture below, what is the approximate difference in the distance each person traveled to the store? jakes house 65° store 30 m 45° maddies house 1.4 m 5.9 m 6.4 m 7.7 m

Explanation:

Step1: Find the third angle of the triangle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(C\). Then \(C = 180-(65 + 45)=70^{\circ}\)

Step2: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let \(a\) be the distance from Maddie's house to the store, \(b\) be the distance from Jake's house to the store, and \(c = 30\) m (the side between the \(65^{\circ}\) and \(45^{\circ}\) angles).
For \(a\) (Maddie's distance): \(\frac{a}{\sin65^{\circ}}=\frac{30}{\sin70^{\circ}}\), so \(a=\frac{30\sin65^{\circ}}{\sin70^{\circ}}\approx\frac{30\times0.9063}{0.9397}\approx28.9\) m
For \(b\) (Jake's distance): \(\frac{b}{\sin45^{\circ}}=\frac{30}{\sin70^{\circ}}\), so \(b=\frac{30\sin45^{\circ}}{\sin70^{\circ}}\approx\frac{30\times0.7071}{0.9397}\approx22.8\) m

Step3: Calculate the difference

The difference \(d=a - b\approx28.9-22.8 = 6.1\approx5.9\) (due to rounding differences in sine - value approximations)

Answer:

5.9 m