A plane flies 1.7 hours at $110 \mathrm{mph}$ on a bearing of $40^{\circ}$. It then turns and flies 2.0 hours at the same speed on a bearing of $130^{\circ}$. How far is the plane from its starting point? The plane is miles from its starting point. (Round to the nearest whole number.)

Step 1 ：First, calculate the distance the plane flew in each leg of its journey. This can be done by multiplying the speed of the plane by the time it flew in each leg. The speed of the plane is \(110 \mathrm{mph}\). The time it flew in the first leg is \(1.7 \mathrm{hours}\), and the time it flew in the second leg is \(2.0 \mathrm{hours}\). Therefore, the distance it flew in the first leg is \(110 \times 1.7 = 187 \mathrm{miles}\), and the distance it flew in the second leg is \(110 \times 2.0 = 220 \mathrm{miles}\).

Step 2 ：Next, calculate the included angle. The difference in the bearings of the two legs is \(130^{\circ} - 40^{\circ} = 90^{\circ}\).

Step 3 ：Finally, use the law of cosines to calculate the distance between the starting point and the ending point of the plane's journey. The law of cosines states that for any triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of those two sides times the cosine of the included angle. In this case, the two sides of the triangle are the distances the plane flew in each leg of its journey, and the included angle is the difference in the bearings of the two legs. Therefore, the distance is \(\sqrt{187^2 + 220^2 - 2 \times 187 \times 220 \times \cos(90^{\circ})} = 289 \mathrm{miles}\).

Step 4 ：Final Answer: The plane is \(\boxed{289}\) miles from its starting point.