### Math Notes

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#### Algebra Solutions

##### Topics || Problems

An airplane flew 660 miles with the wind, and then took 40 minutes longer for the return flight against the wind. If the plane flies 200 mph in still air, find the speed of the wind.

Let $$V$$, $$V_a$$ and $$V_w$$ be the total speed, speed of airplane on still air and speed of the wind respectively .

Let $$t$$ be the time $$(hrs)$$ it took the plane to travel 660 miles, thus the time it took to travel againts the wind is $$t + \frac{2}{3}$$.

$$V = \frac{distance}{time}$$

With the wind: $$V = V_a + Vw$$

Againts the wind: $$V = V_a - V_w$$

$$\frac{660}{t} = 200 + Vw$$

$$\frac{660}{t+\frac{2}{3}} = 200 - Vw$$

$$\frac{660}{t} + \frac{660}{t+\frac{2}{3}} = 400 +0$$

$$660(t+\frac{2}{3} + t) = 400t(t+\frac{2}{3})$$

$$\frac{33}{20}(2t + \frac{2}{3}) = t^2 + \frac{2t}{3}$$

$$\frac{33t}{10} + \frac{11}{10} - t^2- \frac{2t}{3} = 0$$

$$\frac{79t}{30}-t^2 +\frac{11}{10} = 0$$

$$30t^2 -79t -33 = 0$$

$$x_1 = \frac{79+\sqrt{79^2 - 4(30)(-33)}}{60}$$

$$x_1 = 3 ~hrs$$

$$x_2 = \frac{79-\sqrt{79^2 - 4(30)(-33)}}{60}$$

$$x_2 = -0.37$$

Consider the positve root and solve for the speed of the wind.

$$\frac{660}{3} = 200 + V_w$$

$$V_w = 20 ~mph$$ Answer