suppose that we are standing on a bridge 30 feet above a river watching a log floating toward we. if the angle with horizontal to the front of the log is 16.7degree and angle with horizontal to the back of the log is 14degree, how long is the log?
If the front of the log is at distance d, and the log has length x, then
30/d = tan 16.7°
30/(d+x) = tan 14°
eliminate d and we have
30/tan16.7° = 30/tan14° - x
now just solve for x, the length of the log.
-0.06
To find the length of the log, we can use the concept of trigonometry. Let's break down the problem step-by-step:
Step 1: Draw a diagram
- Draw a horizontal line to represent the surface of the river.
- Draw a vertical line to represent the bridge.
- Mark a point on the horizontal line to represent the front of the log.
- Mark another point on the horizontal line to represent the back of the log.
- Connect these two points with a line segment to represent the log.
Step 2: Define the given information
- The height of the bridge from the river surface is 30 feet.
- The angle between the line connecting the front of the log to the point on the bridge and the horizontal is 16.7 degrees.
- The angle between the line connecting the back of the log to the point on the bridge and the horizontal is 14 degrees.
Step 3: Identify relevant trigonometric functions
- We can use the tangent function to solve this problem, as we have the values of angles and the opposite side (vertical height) of the triangle.
- The tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
Step 4: Set up the trigonometric equations
- Using the tangent function, we have:
tan(16.7 degrees) = height of the bridge / length of the log (front side)
tan(14 degrees) = height of the bridge / length of the log (back side)
Step 5: Solve the equations
- Rearrange the equations to solve for the length of the log:
length of the log (front side) = height of the bridge / tan(16.7 degrees)
length of the log (back side) = height of the bridge / tan(14 degrees)
- Substitute the given values:
length of the log (front side) = 30 feet / tan(16.7 degrees)
length of the log (back side) = 30 feet / tan(14 degrees)
Step 6: Calculate the length of the log
- Use a calculator to find the length of the log (front side) ≈ 113.48 feet
- Use a calculator to find the length of the log (back side) ≈ 125.62 feet
Step 7: Summarize the answer
- The length of the log is approximately 113.48 feet from the front and 125.62 feet from the back.
To find the length of the log, we can use the concept of trigonometry.
Let's assume the length of the log is represented by 'L'.
From the given information, we can draw a diagram:
___________________________
/ \
/ \
/ \
/ \
/ \
A B
Log River
O O
|<-------- d1 ----------->|<-------- d2 ----------->|
In this diagram, A and B represent the viewing points from the bridge. O represents the log floating in the river. 'd1' is the distance from Point A to the log, and 'd2' is the distance from Point B to the log.
We know that tan(angle) = opposite/adjacent.
For angle A, tan(16.7 degrees) = (height of the bridge) / d1.
For angle B, tan(14 degrees) = (height of the bridge) / d2.
Let's substitute the values:
tan(16.7 degrees) = 30 / d1,
tan(14 degrees) = 30 / d2.
To find 'd1' and 'd2', we can rearrange the equations:
d1 = 30 / tan(16.7 degrees),
d2 = 30 / tan(14 degrees).
We can then find the total distance traveled by the log, which is given by:
Total distance = d1 + d2.
Finally, we can find the length of the log by subtracting the total distance from the distance between the two viewing points (A and B) on the bridge:
Length of the log = Total distance - (Distance between the viewing points A and B on the bridge).
Please note that the given information assumes the bridge to be perfectly straight, and the log to be floating in a straight line.