Chapter 6.6, Problem 49E

Flying Kites A boy is flying two kites at the same time. He has 380 ft of line out to one kite and 420 ft to the other. He estimates the angle between the two lines to be 30°. Approximate the distance between the kites.

To determine

To find: The approximate distance between the kites.

Answer to Problem 49E

The approximate distance between the kites is $211\text{ft}$.

Explanation of Solution

Formula used:

Law of Cosines:

$a^2=b^2+c^2-2bc\cos A$

Calculation:

Sketch the triangle as follows,

Draw a line between between two kites,

Substitute $a=x$, $b=420$, $c=380$ for $c$ and $A=30°$ in $a^2=b^2+c^2-2bc\cos A$,

$\begin{array}{c}{x}^2={\left(420\right)}^2+{\left(380\right)}^2-2\left(420\right)\left(380\right)\cos 30°\\ =176400+144400-319200\left(0.866\right)\\ =320800-276435.30866\\ =44364.69114\end{array}$

Simplify further,

$\begin{array}{c}\sqrt{x^2}=\sqrt{44364.69114}\\ x=210.629274\\ x\approx 211\text{ft}\end{array}$

Thus, the approximate distance between the kites is $211\text{ft}$.