Question 1

The height of the tree is 32 feet.

To find the height of the tree when the Sun is 35 degrees above the horizon and the shadow is 46 feet long, we can use the tangent function in trigonometry. The tangent of the angle (35 degrees) is equal to the height of the tree divided by the length of the shadow, leading us to conclude that the tree's height is approximately 32 feet.

A) 32 feet

This option is correct. Using the tangent function, we calculate the height of the tree as follows: height = tan(35 degrees) × 46 feet. This calculation yields approximately 32 feet, making this the accurate representation of the tree's height.

B) 56 feet

This option is incorrect. A height of 56 feet would imply a much steeper angle than 35 degrees when the shadow length is 46 feet. The tangent of 35 degrees does not support a height that high given the shadow length.

C) 66 feet

This option is also incorrect. Similarly to Option B, a height of 66 feet would suggest an unrealistic slope for the given shadow length at an angle of 35 degrees. The calculations indicate that the tree cannot be this tall.

D) 80 feet

This option is incorrect as well. A tree height of 80 feet would require an even more extreme angle than 35 degrees when the shadow is only 46 feet long. Hence, this option does not align with the trigonometric calculations.

Conclusion

The correct answer is 32 feet as it accurately reflects the calculations derived from the tangent function for the given angle and shadow length. All other options fail because they suggest heights that do not conform to the geometric relationship between angle, height, and shadow length at 35 degrees.