A bird at the top of a tree looks down on a field mouse at an angle of depression of 65°. If the field mouse is 30 meters from the base of the tree, find the vertical distance from the ground to the bird’s eyes. Round the answer to the nearest tenth.
Review your basic trig functions.
As always, draw a diagram.
If the distance is x,
tan65° = x/30
so it is 30/tan65 degress plug it in a claculator and you should get 13.9 which equals 14.
To find the vertical distance from the ground to the bird's eyes, we can use trigonometric functions.
First, let's draw a diagram to visualize the problem. We have a right triangle formed by the bird, the tree, and the ground.
The angle of depression is the angle formed between the bird's line of sight (looking down) and the horizontal line. In this case, the angle of depression is given as 65°.
The distance from the base of the tree to the field mouse is 30 meters, which is the horizontal distance.
Let's define the vertical distance from the ground to the bird's eyes as "x".
Using trigonometry, we can use the tangent function to relate the angle of depression, the vertical distance, and the horizontal distance:
tan(angle) = opposite / adjacent
In this case, the angle is the angle of depression (65°), the opposite side is the vertical distance "x", and the adjacent side is the horizontal distance (30 meters).
tan(65°) = x / 30
To find the vertical distance "x", rearrange the equation:
x = 30 * tan(65°)
Using a calculator, evaluate the right side of the equation:
x ≈ 30 * 2.1445
x ≈ 64.3
Therefore, the vertical distance from the ground to the bird's eyes is approximately 64.3 meters (rounded to the nearest tenth).