To find the width of a river, a surveyor set up his surveying equipment
at C on one bank and sighted across to a point B on the opposite bank;
then, turning through an angle of 90deg, he laid off a distance CA=225
m. Finally, setting the equipment at A, he measured angle CAB as
48°20”. Find the width of the river. Provide an Illustration
I'll do this one. How about you make a stab at the others?
In every case, draw a diagram, and think of your basic trig functions. Show your work if you get stuck.
w/225 = tan48°20”
now finish it off
Well, if the surveyor turned through an angle of 90 degrees, it seems like he really didn't want to get wet and preferred a nice view of the river from the comfort of the riverbank.
Anyway, let's get to the math. From the given information, we have CA = 225 m and angle CAB = 48°20". We're trying to find the width of the river, which we'll call x.
Now, if we draw a diagram, we can see that triangle CAB is a right triangle with angle CAB as the right angle. We know the measure of angle CAB, so we can use some trigonometry to find the width of the river.
In particular, we can use the tangent function. The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side. In this case, the side opposite angle CAB is the width of the river (x), and the side adjacent to angle CAB is CA (225 m). So we can write:
tan(CAB) = x / CA
Now we can substitute the known values:
tan(48°20") = x / 225
To find x, we can multiply both sides of the equation by 225:
x = 225 * tan(48°20")
Now, let me get my calculator... *beep boop beep* Okay, according to my calculations, the width of the river is approximately 248.3 meters.
I hope that helps! And remember, always bring your clown nose when surveying rivers. It adds a certain je ne sais clown.
To find the width of the river, we can use the concept of trigonometry and the given information.
Let's denote the width of the river as "x".
According to the given information, the surveyor set up the equipment at point C on one bank and sighted across to point B on the opposite bank. He then turned through an angle of 90 degrees and laid off a distance CA = 225 m.
First, let's draw a diagram to visualize the situation:
```
C
|\
| \
| \
x | \ B
| \
| \
A------D
```
In the diagram, A, B, C, and D represent the points described in the problem. The line segment AB represents the width of the river, and AC represents the distance laid off by the surveyor.
We need to find the width of the river, which is AB.
From the given information, we know that angle CAB is 48°20" and angle CAD is 90°. We can use the trigonometric function tangent (tan) to determine the width of the river.
The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
In this case, tan(CAB) = AB / AC.
Let's convert the angle CAB to decimal degrees:
48°20" = 48 + 20/60 = 48.3333°
Now we can calculate the width of the river:
tan(48.3333°) = AB / 225
To solve for AB, we rearrange the equation and substitute the values:
AB = tan(48.3333°) * 225
Using a calculator, we find:
AB ≈ 277.73 meters (rounded to 2 decimal places)
Therefore, the approximate width of the river is 277.73 meters.
To find the width of the river, we can use trigonometry and the information provided in the problem. Here's how you can approach this problem step by step:
Step 1: Analyze the given information and draw an illustration. Let's label the points as shown in the figure below:
B
/|
/ |
/ |
/ |x (width of the river)
A /____| C
The point A represents the surveyor's position, point B represents the opposite bank, and point C represents the initial surveying position.
Step 2: Use the information given to find the length of line segment AC. It is given that CA = 225 m.
Step 3: Use trigonometry to find the length of line segment BC. For this, we will use the tangent function.
tan(CAB) = BC / AC
Rearranging the equation, we have:
BC = AC * tan(CAB)
Substitute the given values:
BC = 225 * tan(48°20”)
Before calculating the value, it's important to convert degrees and minutes to decimal form. To convert minutes to degrees, divide the number of minutes by 60:
CAB in decimal form = 48 + (20/60) = 48.3333 degrees
Now, calculate the value of tan(48.3333) using a scientific calculator or any online calculator.
Step 4: Calculate the width "x" of the river. The width of the river is the sum of line segments AB and BC:
x = AB + BC
Assuming AB is negligible (as a direct line of sight from C to B), we can approximate AB to be zero for practical purposes. Therefore:
x ≈ BC
Substitute the value of BC calculated in the previous step.
Step 5: Round your final answer to an appropriate number of decimal places based on the level of precision given in the problem (usually guided by the given values and any instructions provided).
Illustration:
Here is an illustration of the problem setup:
B
/|
/ |
/ |
/ |225 m
A /____| C
This illustration represents point C on one bank, point A on the other bank, and the width of the river denoted as "x."
By following these steps, you should be able to find the width of the river.