A forest ranger spots a fire from a 28- foot tower.The angle of depression from the tower to the fire is 11.To the nearest foot,how far is the fire from the base of the tower?Show the steps you use to find the solution.
Draw a diagram
cot 11° = d/28
d=144 ft
To find the distance from the base of the tower to the fire, we can use trigonometry and the concept of angle of depression.
Step 1: Draw a diagram:
Start by drawing a diagram that represents the situation described in the problem. Draw a vertical line to represent the tower and label it with its height, which is 28 feet. Then draw a horizontal line from the base of the tower to represent the distance to the fire. Lastly, label the angle of depression, which is 11 degrees. This diagram will help visualize the problem and make it easier to solve.
Step 2: Identify the right triangle:
Notice that we have a right triangle formed by the tower, the distance to the fire, and a horizontal line parallel to the ground.
Step 3: Identify the trigonometric ratio:
Since we have the angle of depression (11 degrees) and we are looking for the distance from the base of the tower to the fire, we can use the tangent ratio, which is defined as the opposite side divided by the adjacent side. In this case, the opposite side is the height of the tower (28 feet) and the adjacent side is the distance to the fire, which we want to find.
Step 4: Set up and solve the equation:
Using the tangent ratio: tan(angle) = opposite side / adjacent side
In this case: tan(11) = 28 / x (where x is the distance to the fire)
Step 5: Solve for x:
To solve for x, we need to rearrange the equation by multiplying both sides by x:
x * tan(11) = 28
Then, divide both sides by tan(11):
x = 28 / tan(11)
Using a scientific calculator, calculate tan(11) to get the value of the tangent of 11 degrees. Then divide 28 by this value to get the distance to the fire, which is approximately 148.24 feet.
To the nearest foot, the distance from the base of the tower to the fire is 148 feet.