Final answer:
The distance CD between two boats can be found using trigonometry for a 30-60-90 right triangle. Since none of the provided options correspond with the calculated distance of approximately 43.3 feet, there may be an error in the options or missing information that affects the calculation.
Step-by-step explanation:
To find the distance CD between the boats, we first need to calculate the lengths of AC and BC in triangle ABC, where angle ABC is a right angle and angle ACB is 60 degrees. Since AB is the side opposite the right angle, it is the hypotenuse of the right triangle ABC. Using trigonometry, we find that in a 30-60-90 right triangle, the side opposite the 60-degree angle (AC) is equal to the hypotenuse (AB) multiplied by the square root of 3 divided by 2, and the side opposite the 30-degree angle (BC) is equal to half of the hypotenuse. Therefore, AC = (100 feet) * (sqrt(3) / 2) and BC = (100 feet) / 2.
Knowing that triangle ADC is also a 30-60-90 right triangle, we can calculate the length of AD using the same ratios, with CD being the side opposite to the 30-degree angle. Thus AD is twice the length of CD. Since AC = AD, the length of CD is half the length of AC. Therefore, CD = AC / 2 = ((100 * sqrt(3)) / 2) / 2 = (100 * sqrt(3)) / 4. Calculating this value gives us CD ≈ 43.3 feet.
The correct answer from the provided choices depends on rounding policy, but none of the given options correctly match our calculated value. It's possible there may have been an error in the given options, or there may have been additional information such as the length of AD that needs to be considered.