Final answer:
To find the height of the tree, the trigonometric tangent function is used with the given 30° angle and the 24 feet shadow length. Solving tan(30°) = h / 24 yields a tree height of approximately 13.848 feet. Since 14 feet is not an option, the closest answer is 12 feet.
Step-by-step explanation:
The problem involves finding the height of a tree using trigonometry, specifically the tangent function. Given that the tree casts a shadow 24 feet long when the sun's rays form a 30° angle with the earth, we use the trigonometric equation tan(θ) = opposite/adjacent, where the opposite side is the tree's height that we are solving for (h), and the adjacent side is the length of the shadow (24 feet). In this case, θ is given as 30°.
Therefore, tan(30°) = h / 24. We know that tan(30°) equals approximately √3/3 or about 0.577. So 0.577 = h / 24. To find h, multiply both sides by 24 to get h = 24 * 0.577 = 13.848 feet. Rounding to the nearest whole number, the tree is approximately 14 feet tall. However, since this option is not given, the closest answer would be B) 12 feet.