Question

a beach has two floating docks. one is 650 meters east of the lifeguard stand. the other is 60° southeast and 750 meters from the lifeguard stand. law of cosines: a² = b² + c² - 2bc cos(a) rounded to the nearest meter, what is the distance between the docks? round to the nearest meter. 589 meters 705 meters 792 meters 861 meters

Explanation:

Step1: Identify values for law of cosines

Let $b = 650$, $c = 750$, and $A=60^{\circ}$. The law of cosines is $a^{2}=b^{2}+c^{2}-2bc\cos(A)$.

Step2: Substitute values into formula

$a^{2}=650^{2}+750^{2}-2\times650\times750\times\cos(60^{\circ})$.
First, calculate the squares and the cosine - related part:
$650^{2}=422500$, $750^{2}=562500$, and $\cos(60^{\circ})=\frac{1}{2}$.
$2\times650\times750\times\cos(60^{\circ})=2\times650\times750\times\frac{1}{2}=650\times750 = 487500$.
Then $a^{2}=422500 + 562500-487500$.

Step3: Simplify the right - hand side

$a^{2}=422500+562500 - 487500=497500$.

Step4: Solve for $a$

$a=\sqrt{497500}\approx705.337$.
Rounding to the nearest meter, $a\approx705$.

Answer:

705 meters