A forest ranger in a 140-foot observation tower sees a fire moving in a direct path toward the windmill by the lake. The angle of depression to the fire is 3°, and the angle of depression to the windmill is 8°. To the nearest foot, how close is the fire to the windmill?
tan8 = 140/X1.
X1 = 140/tan8 = 996.2 Ft.
tan3 = 140/(996+X2)
996+X2 = 140/tan3 = 2671
X2 = 2671-996 = 1675 Ft. = Distance from
fire to windmill.
1675 ft
1675 feet. Gosh im so smart
1675 ft. Im smarter tho
To solve this problem, we can use trigonometry and specifically the tangent function. Here's how we can find the distance between the fire and the windmill:
1. Let's start by drawing a diagram. Place the observation tower at the top and mark the fire and the windmill on the ground level.
2. Label the height of the observation tower as 140 ft. Also, label the distance between the observation tower and the fire as "x" and the distance between the observation tower and the windmill as "y".
3. From the information given, we know that the angle of depression to the fire is 3°. This means that the angle between the horizontal line from the observer to the fire and the line of sight to the fire is 3°.
4. Similarly, the angle of depression to the windmill is 8°. Again, this means that the angle between the horizontal line from the observer to the windmill and the line of sight to the windmill is 8°.
5. By using trigonometry, we can determine the relationships between these angles and the distances involved. The tangent function is useful in this case because it relates the angle to the ratio of the opposite side to the adjacent side.
6. For the angle of depression to the fire, we have the ratio tan(3°) = opposite side (140 ft) / x. Rearranging the equation, we get x = 140 ft / tan(3°).
7. Similarly, for the angle of depression to the windmill, we have the ratio tan(8°) = opposite side (140 ft) / y. Rearranging the equation, we get y = 140 ft / tan(8°).
8. Now, we can substitute the values of x and y into the equation y - x to find the distance between the fire and the windmill.
9. Simplifying the equation, we have y - x = (140 ft / tan(8°)) - (140 ft / tan(3°)).
10. Plugging the values into the equation, we can calculate y - x ≈ 6821.68 ft.
Therefore, to the nearest foot, the fire is approximately 6822 feet close to the windmill.