Question

1. madi rides her bike east for 7 miles and then rotates around her starting point 40 degrees and finally traveled directly back to her starting point what is her total distance traveled? round to nearest whole number.

Explanation:

Step1: Identify the triangle type

Madi's path forms an isosceles triangle with two sides equal (the two legs from the starting point) and the included angle of \(40^\circ\). Let the length of the first leg (east) be \(a = 7\) miles, the second leg (after rotation) be \(b = 7\) miles, and the included angle \(\theta=40^\circ\). We need to find the length of the third side \(c\) (the return trip) using the Law of Cosines: \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\)

Step2: Calculate the third side

Substitute \(a = 7\), \(b = 7\), and \(\theta = 40^\circ\) into the formula. First, calculate \(\cos(40^\circ)\approx0.7660\)
\[

$$\begin{align*} c^{2}&=7^{2}+7^{2}-2\times7\times7\times\cos(40^\circ)\\ &=49 + 49-98\times0.7660\\ &=98-75.068\\ &=22.932 \end{align*}$$

\]
Then take the square root: \(c=\sqrt{22.932}\approx4.79\) miles

Step3: Calculate total distance

Total distance \(D=a + b + c=7 + 7+4.79 = 18.79\approx19\) miles (rounded to nearest whole number)

Answer:

19