A surveyor measures the distance across a
straight river by the following methods: Starting directly across from a tree on the opposite
bank, she walks 249 m along the river bank to
establish a baseline. Then she sights across to
the tree. The angle from her baseline to the
tree is 38.2
How wide is the river?
tan 38.2 = d /249
d = 249 tan 38.2 = 196 m
To find the width of the river, we can use trigonometry. We will use the concept of tangent of an angle.
Tangent is defined as the ratio of the opposite side to the adjacent side of a right triangle. In this case, the opposite side is the width of the river, and the adjacent side is the baseline distance measured along the river bank.
Let's label the width of the river as "x". We can set up the following equation using the tangent function:
tan(38.2) = x / 249
To solve for x, we can multiply both sides of the equation by 249:
x = 249 * tan(38.2)
Using a scientific calculator, we can find the value of tangent(38.2) to be approximately 0.788.
x ≈ 249 * 0.788 ≈ 196.212
Therefore, the width of the river is approximately 196.212 meters.
To find the width of the river, we can use trigonometry and the given information. Here's how you can calculate it step-by-step:
Step 1: Draw a diagram to visualize the situation described in the problem. Label the baseline as "249 m" and the angle as "38.2°".
Step 2: Identify the trigonometric function that relates the known values of the triangle to the unknown value (width of the river). In this case, we know the length of the adjacent side (the baseline) and the measure of the adjacent angle (38.2°). The trigonometric function that relates these values is the cosine function.
Step 3: Write the cosine equation using the known values and the unknown value. In this case, we want to find the width of the river (let's call it "x"), so the equation will be:
cos(38.2°) = adjacent / hypotenuse
Step 4: Substitute the known values into the equation:
cos(38.2°) = 249 / x
Step 5: Solve the equation for x by isolating it on one side:
x * cos(38.2°) = 249
Step 6: Divide both sides of the equation by cos(38.2°):
x = 249 / cos(38.2°)
Step 7: Use a calculator to evaluate the expression:
x ≈ 301.35 m
Therefore, the width of the river is approximately 301.35 meters.