A surveyor measures the distance across a straight river by the following method. Starting directly across from a tree on the opposite bank, she walks 90 m along the riverbank to establish a baseline. Then she sights across to the tree. The angle from her baseline to the tree is 50.0°. How wide is the river?
Draw a right-triangle with an angle of 50° to the base which measures b=90 m.
The height of the triangle is the width of the river, w.
so w=b*tan(50°)
45
To find the width of the river, we can use trigonometry. Let's go step by step.
Step 1: Draw a diagram to visualize the problem. This will help us understand the given information better.
Let's assume the tree is point T, the starting point on the riverbank is point A, and the endpoint of the baseline on the riverbank is point B. The river forms a straight line between points A and T.
A ----------- B
\ /
\ /
T \ /
\ /
\ /
C (river width)
Step 2: Identify the given information:
- The distance from point A to point B (the baseline) is 90 m.
- The angle formed between the baseline AB and the line of sight to the tree (angle ACT) is 50.0°.
Step 3: Use trigonometry to find the width of the river.
We can use the tangent function to calculate the width of the river (BC).
tan(50.0°) = BC / AB
Step 4: Substitute the values into the equation.
tan(50.0°) = BC / 90m
Step 5: Solve for BC.
Multiply both sides of the equation by 90m to isolate BC.
BC = tan(50.0°) * 90m
Step 6: Calculate the width of the river.
Using a calculator, find the value of tan(50.0°) and multiply it by 90m to obtain the width of the river.
BC ≈ 90m * 1.1918 ≈ 107.27m
Therefore, the width of the river is approximately 107.27 meters.
To find the width of the river, we can use trigonometry and the information given. Let's break down the problem into smaller steps:
Step 1: Draw a diagram of the situation to visualize it.
Imagine a straight river with a tree on the opposite bank. The surveyor starts directly across from the tree, walks 90 m along the riverbank to establish a baseline, and sights across to the tree.
Step 2: Identify the relevant information given in the problem:
- The surveyor walks 90 m along the river bank (this is the baseline).
- The angle from the baseline to the tree is 50.0°.
Step 3: Identify the trigonometric functions that can help solve the problem:
In this case, we can use the tangent function since we have the length of the baseline and the measure of an angle. The tangent function is:
tan(angle) = opposite/adjacent
Step 4: Apply the tangent function to find the width of the river:
Let's define the width of the river as 'x'.
tan(50°) = opposite (x) / adjacent (90m)
tan(50°) = x / 90
Solve this equation for 'x' by multiplying both sides by 90:
x = 90 * tan(50°)
x ≈ 86.36 meters
So, the width of the river is approximately 86.36 meters.