A biologist wants to know the width w of a river to properly set instruments for an experiment. From point A, the biologist walks downstream 100 feet and sights to point C (see figure). From this sighting, it is determined that

Question

A biologist wants to know the width w of a river to properly set instruments for an experiment. From point A, the biologist walks downstream 100 feet and sights to point C (see figure). From this sighting, it is determined that

𝜃 = 50°.
How wide is the river? (Round your answer to one decimal

Answer 1

you don't say how 𝜃 is measured, but if C is directly across from A, and the width is w, then

w/100 = tan𝜃
or
w/100 = cot𝜃

Answer 2

To find the width of the river, we can use trigonometry and the information given in the problem.

Let's break down the problem step by step:

Step 1: Identify the relevant triangle
In this case, we have a triangle formed by points A, B, and C. The width of the river (w) is the length of BC.

Step 2: Find the length of AC
From the problem statement, we know that the biologist walks downstream 100 feet from point A to point C. So, the length of AC is 100 feet.

Step 3: Use trigonometry to find the width of the river
We have an angle 𝜃 (theta) given in the problem, which is the angle between AB and AC. To find the width of the river, we can use the tangent function.

The tangent function is defined as:

tan(𝜃) = opposite / adjacent

In our case:
opposite = BC (width of the river)
adjacent = AC (length of AC)

So, we can rewrite the equation as:

tan(𝜃) = BC / AC

Substituting the known values:

tan(50°) = BC / 100 feet

Now, we need to solve for BC (width of the river). Rearrange the equation:

BC = tan(50°) * 100 feet

Use a scientific calculator or a trigonometric table to calculate the value of tan(50°), then multiply it by 100 feet to get the width of the river.

Finally, round your answer to one decimal place as requested in the problem statement.