To solve the problem of finding the distance between the two floating docks, we can use the law of cosines. Here's a step-by-step explanation of how we apply it:
1. Understand the given values:
- The distance from the lifeguard stand to the first dock [tex]\(b\)[/tex] is 650 meters.
- The distance from the lifeguard stand to the second dock [tex]\(c\)[/tex] is 750 meters.
- The angle between the two distances, [tex]\(A\)[/tex], is [tex]\(60^\circ\)[/tex].
2. Use the law of cosines to find the distance [tex]\(a\)[/tex] between the two docks:
The law of cosines formula is:
[tex]\[
a^2 = b^2 + c^2 - 2bc \cos(A)
\][/tex]
3. Convert the angle from degrees to radians.
- Since the angle [tex]\(A\)[/tex] is given in degrees, we need to convert it to radians because the cosine function in trigonometry usually uses radians.
- The conversion formula is:
[tex]\[
A_{\text{radians}} = A \times \frac{\pi}{180}
\][/tex]
4. Substitute the given values into the formula:
Using [tex]\(b = 650\)[/tex] meters, [tex]\(c = 750\)[/tex] meters, and [tex]\(A = 60^\circ\)[/tex]:
[tex]\[
A_{\text{radians}} = 60 \times \frac{\pi}{180} = \frac{\pi}{3}
\][/tex]
5. Calculate the cosine of [tex]\(A_{\text{radians}}\)[/tex]:
[tex]\[
\cos(\frac{\pi}{3}) = 0.5
\][/tex]
6. Apply the cosine value into the law of cosines formula:
[tex]\[
a^2 = 650^2 + 750^2 - 2 \cdot 650 \cdot 750 \cdot 0.5
\][/tex]
7. Compute the squares and product:
[tex]\[
a^2 = 422500 + 562500 - 2 \cdot 650 \cdot 750 \cdot 0.5
= 422500 + 562500 - 487500
= 497500
\][/tex]
8. Take the square root of both sides to find [tex]\(a\)[/tex]:
[tex]\[
a = \sqrt{497500} \approx 705.0
\][/tex]
9. Round the result to the nearest meter:
[tex]\(a \approx 705\)[/tex] meters.
So, the distance between the two docks, rounded to the nearest meter, is [tex]\(705\)[/tex] meters. The correct answer is 705 meters.
