4. An observer is near a river and wants to calculate the distance across the river. He measures the angle between his observations of two points on the shore, one on his side and one on the other side, to be 28º. The distance between him and the point on his side of the river can be measured and is 300 feet. The angle formed by him, the point on his side of the river, and the point directly on the opposite side of the river is 128º. What is the distance across the river?
Assuming that the first-named point on the opposite side (P) is the point directly opposite from the point on his own side (Q), then we have, if the observer is point O,
m<POQ = 28
m<OQP = 128, so
m<OPQ = 24
Now, using the law of sines, if the width is w, we have
w/sin28 = 300/sin24
346.27
To solve this problem, we will use the trigonometric concept of tangent.
Let's start with the given information:
Angle between the observer, the point on his side, and the point on the other side of the river = 128º.
Distance between the observer and the point on his side of the river = 300 feet.
Angle between the observer, the point on his side, and the point on the other side of the river = 28º.
We will use the angle of 28º and the distance of 300 feet to calculate the distance across the river.
Step 1: Draw a diagram to visualize the problem.
O
|\
| \
300\ \
| \
|28°\
-----
R
O: Observer
R: Point on the other side of the river
Angle marked as 28° is between the observer, the point on his side and the point on the other side of the river.
The distance between the observer and the point on his side = 300 feet.
Step 2: Use the tangent function (tan) to calculate the distance across the river.
We have the angle of 28° and the distance of 300 feet. The tangent of an angle is the ratio of the opposite side to the adjacent side.
Opposite Side
tan(28°) = ---------------------
Adjacent Side
Let x be the distance across the river.
x
tan(28°) = --------------
300
Step 3: Rearrange the equation to solve for x.
Multiply both sides of the equation by 300 to isolate x:
x = 300 * tan(28°)
Step 4: Calculate the distance across the river.
Using a calculator:
x ≈ 171.71 feet
Therefore, the distance across the river is approximately 171.71 feet.
To solve this problem, we can use the concept of trigonometry and specifically the properties of right triangles.
Let's denote the distance across the river as "d" (the quantity we need to find).
From the problem statement, we know the following:
- The observer measures the angle between two points on the shore near him, one on his side and one on the other side, to be 28 degrees.
- The distance between the observer and the point on his side of the river is 300 feet.
- The angle formed by the observer, the point on his side of the river, and the point directly on the opposite side of the river is 128 degrees.
Based on this information, we can visualize the situation and construct a triangle:
|\
| \
d | \ 300 feet
| \
| \
|_____\
Now, let's analyze the triangle and use trigonometry to find the distance across the river (d).
1. First, let's focus on the triangle formed by the observer, the point on his side of the river, and the point directly on the opposite side.
Since we know the angle between the observer, the point on his side, and the point directly across the river is 128 degrees, we can use the trigonometric function cosine (cos) to solve for the length of the side opposite to this angle.
cos(128) = (300 feet) / x, where x denotes the length of the side opposite the angle.
Solving this equation for x, we get:
x = (300 feet) / cos(128).
2. Now we have the length of the side opposite the angle of 128 degrees. Since we want to find the length of the side across the river (d), we need to consider the angle of 28 degrees between the observer and the point on the other shore.
By using the trigonometric function sine (sin), we can set up the following relationship:
sin(28) = (x) / d, where d denotes the length of the side across the river.
Rearranging the equation to solve for d, we get:
d = x / sin(28).
3. Now we can substitute the expression for x that we found in step 1 into the equation from step 2:
d = [(300 feet) / cos(128)] / sin(28).
Using a calculator, we can evaluate this expression to find the distance across the river (d).