Final answer:
Using trigonometric ratios for the given angles of 30° and 60°, the only possible shadow length provided in the options, when Ben checks his sundial, is option B, 2√3 inches. The cosine function is used to find the length of the shadow based on the height of the stick and the angle of the sun's rays.
Step-by-step explanation:
To determine the possible lengths of the shadow when Ben checks the sundial, we need to consider the angles at which the sun's rays hit the stick and the resulting triangles formed. If the angle between the sun's rays and the ground is 30°, we have a right-angled triangle where the stick is the opposite side (height), the shadow is the adjacent side, and the sun's rays form the hypotenuse. We can use trigonometric ratios to calculate the length of the shadow.
For a 30° angle:
- The trigonometric function to use is the cosine, which is the ratio of the adjacent side to the hypotenuse.
- So, cosine(30°) = (shadow length) / (stick height).
- Since cosine(30°) = √3 / 2, and the stick height is 6 inches, shadow length = (6 inches) × (√3 / 2) = 3√3 inches.
- Therefore, option B (6 inches × √3, simplified to 2√3 inches) is a possible shadow length.
For a 60° angle:
- The trigonometric function to use is the cosine, which is the ratio of the adjacent side to the hypotenuse for this right-angled triangle.
- Since cosine(60°) = 1 / 2, and the stick height is 6 inches, shadow length = (6 inches) × (1 / 2) = 3 inches.
- There is no option matching exactly 3 inches, so we can rule this out.
With the provided options, only option B is a mathematically possible shadow length when considering the angles of 30° and 60° between the sun's rays and the ground.
It's worth mentioning that if a 45° angle was considered, then we would have shadow length = stick height because in a 45° angle triangle the lengths of the sides opposite and adjacent to the angle are equal. However, since this angle was not part of the question, we do not consider any resulting shadow lengths.