The figure below shows a flying kite. At a certain moment, the kite string forms an angle of elevation of $75^\circ$ from point $A$ on the ground. At the same moment, the angle of elevation of the kite at point $B$, $240$ ft from $A$ on level ground, is $45^\circ$. What is the length, in feet, of the string?

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Call the point where the kite is $C$ and opposite side $c=240$. Then, using law of sines,

$$\frac{c}{\sin{C}} = \frac{b}{\sin{B}}$$ $$\frac{240}{\sin{60^\circ}} = \frac{\text{string length}}{\sin{45^\circ}}$$

We can use the calculator, but it will end up with a decimal number. You can check to see if the first two choices match this decimal number. Alternatively, if you know your unit circle,

$$\frac{240}{\sin{60^\circ}} = \frac{\text{string length}}{\sin{45^\circ}}$$ $$\frac{240}{\frac{\sqrt{3}}{2}} = \frac{\text{string length}}{\frac{\sqrt{2}}{2}}$$ $$\frac{240}{\sqrt{3}}= \frac{\text{string length}}{\sqrt{2}}$$ $$\frac{240}{\sqrt{3}}\cdot \sqrt{2} = \text{string length}$$ $$\frac{240\cdot \sqrt{6}}{3} = \text{string length}$$ $$\text{string length} = \boxed{80\sqrt{6}}$$