From the top of a fire tower, a forest ranger sees his partner on the ground at an angle of depression of 40º. If the tower is 45 feet in height, how far is the partner from the base of the tower, to the nearest tenth of a foot? (Detailed explanation please)
To find the distance from the base of the tower to the partner, we can use trigonometry and specifically the tangent function.
Let's break down the information given:
Height of the tower = 45 feet
Angle of depression = 40º
We need to find the distance from the base of the tower to the partner. Let's call this distance "x".
Now, let's construct a right triangle using the given information. The height of the tower is the opposite side of the triangle, and the distance "x" is the adjacent side of the triangle. The angle of depression (40º) is the angle formed between the hypotenuse and the adjacent side.
Using the trigonometric ratio of tangent, we can set up the following equation:
tangent(angle) = opposite / adjacent
In this case, we have:
tangent(40º) = 45 / x
To find "x", we can rearrange the equation:
x = 45 / tangent(40º)
Now, let's calculate it. Firstly, note that trigonometric functions usually operate in radians instead of degrees. We need to convert the angle from degrees to radians. To do that, multiply the angle by π (pi) and divide by 180.
angle in radians = 40º * π / 180 ≈ 0.69813 radians
Now, we can calculate the tangent of the angle in radians:
tangent(0.69813 radians) ≈ 0.8391
Finally, divide the height of the tower by the tangent value to get the distance "x":
x ≈ 45 / 0.8391 ≈ 53.7 feet
Therefore, the partner is approximately 53.7 feet away from the base of the tower, rounded to the nearest tenth of a foot.
To solve this problem, we can use trigonometry and specifically, tangent function.
Let's visualize the situation. We have a fire tower, and the forest ranger is standing at the top of it. The partner is on the ground, forming an angle of depression of 40º with the ranger's line of sight.
We are given that the height of the tower is 45 feet. Let's call the distance between the partner and the base of the tower "x" (in feet). We need to find the value of "x".
Remember that the tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the tower (45 feet) and the adjacent side is the distance between the partner and the base of the tower (x feet).
Using the tangent function, we can set up the following equation:
tan(40º) = opposite/adjacent
tan(40º) = 45/x
Now we can solve for "x" by rearranging the equation:
x = 45 / tan(40º)
Calculating this using a scientific calculator, we find:
x ≈ 45 / 0.8391 ≈ 53.71 feet
Therefore, the partner is approximately 53.71 feet away from the base of the tower. Rounded to the nearest tenth, the distance is approximately 53.7 feet.
Assuming the ground and horizontal are parallel, the angle from the ground at the partner up toward the ranger on the tower is also 40 degrees.
Now you have a right triangle and the side opposite the 40 degrees at the partner is the 45 foot tower so
tan 40 = 45 / d
d = 45 / tan 40