Question

a plane flies 1.4 hours at 120 mph on a bearing of 20°. it then turns and flies 3.8 hours at the same speed on a bearing of 110°. how far is the plane from its starting point? the plane is miles from its starting point. (round to the nearest whole number.)

Explanation:

Step1: Calculate the first - leg distance

The distance formula is $d = vt$. For the first part of the flight, $v = 120$ mph and $t=1.4$ hours. So, $d_1=120\times1.4 = 168$ miles.

Step2: Calculate the second - leg distance

For the second part of the flight, $v = 120$ mph and $t = 3.8$ hours. So, $d_2=120\times3.8=456$ miles.

Step3: Find the included angle

The first bearing is $20^{\circ}$ and the second is $110^{\circ}$, so the included angle $\theta=110 - 20=90^{\circ}$.

Step4: Use the Pythagorean theorem

Since the triangle formed is a right - triangle (as the included angle is $90^{\circ}$), if the two sides of the right - triangle are $a = 168$ and $b = 456$, and the hypotenuse is $c$ (the distance from the starting point), by the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$.
Substitute $a = 168$ and $b = 456$ into the formula: $c=\sqrt{168^{2}+456^{2}}=\sqrt{28224 + 207936}=\sqrt{236160}\approx486$.

Answer:

486