Two fire wardens are stationed at locations A and B, which are 50 km apart. Each warden sights the forest fire at C. Given that angle CAB is 52 degrees and CBA is 43.0, find the distance from C to the nearer warden, to the nearest 0.1 km
angle C = 85° (180-52-43)
Labeling the sides a,b,c opposite angles A,B,C, you want side b, since angle B is the smallest angle.
b/sinB = c/sinC
b/sin43° = 50/sin85°
Now just solve for b
Well, it seems like the fire wardens are in quite a "fiery" situation, huh? Let's see if we can come up with a "blazing" solution!
First, let's draw a diagram to make things "hotter"... I mean, clearer. We have point A, point B, and point C, forming a triangle. Now, based on the angles you provided, we have an angle of 52 degrees at point A (warden A) and an angle of 43 degrees at point B (warden B).
Now, since we're dealing with a triangle, the three angles add up to 180 degrees. So, the angle at point C is:
180 - 52 - 43 = 85 degrees.
Now, we have a "smoldering" triangle ABC with angles 52, 43, and 85. We're trying to find the distance from point C to the nearer warden, which we'll call distance X.
To find X, we can use the Law of Sines, which states that in any triangle:
a/sin(A) = b/sin(B) = c/sin(C)
In our case, let's use side BC and angle B to find X:
BC/sin(52) = X/sin(85)
Now, we can rearrange the equation to find X:
X = BC * (sin(85) / sin(52))
Given that BC is the distance between the two wardens, which is 50 km, we can substitute it in:
X = 50 * (sin(85) / sin(52))
Calculating this value, we find:
X ≈ 65.8 km
So, the distance from point C to the nearer warden is approximately 65.8 km, to the nearest 0.1 km. We hope the fire wardens can handle the "heat" and extinguish the fire quickly! Stay safe out there!
To find the distance from C to the nearer warden, we can use the Law of Sines.
Let's assume that the distance from C to the nearer warden is x km.
Using the Law of Sines, we have:
sin(CAB) / AC = sin(CBA) / BC
Plugging in the known values:
sin(52°) / AC = sin(43°) / 50 km
Rearranging the equation:
AC = (sin(52°) / sin(43°)) * 50 km
Calculating the value of AC:
AC ≈ (0.7880 / 0.6820) * 50 km
AC ≈ 0.9135 * 50 km
AC ≈ 45.675 km
Since we assumed that the distance from C to the nearer warden is x km, we can subtract x from 45.675 km to find the distance from C to the farther warden:
45.675 km - x km = 50 km - x km
Now, we need to find the value of x that makes these two distances equal:
45.675 km - x = 50 - x
Simplifying the equation:
45.675 km = 50
x = 45.675 km - 50 km
x ≈ 4.325 km
Therefore, the distance from C to the nearer warden is approximately 4.325 km.
To solve this problem, we can make use of the trigonometric functions sine and cosine. Here's how to calculate the distance from C to the nearer warden:
1. Draw a diagram of the situation described. Label the points A, B, and C as mentioned in the question.
2. Note that angles CAB and CBA represent the angles made by the lines AC and BC with the line AB, respectively.
3. Convert the given angles from degrees to radians. To convert an angle from degrees to radians, multiply it by 𝜋/180.
Angle CAB = 52 degrees × 𝜋/180 = 0.9076 radians
Angle CBA = 43 degrees × 𝜋/180 = 0.7505 radians
4. Use the sine and cosine functions to set up two equations:
For the triangle ABC, we have:
cos(52) = x / 50, where x represents the distance from A to C.
For the triangle BAC, we have:
cos(43) = x / 50, where x represents the distance from B to C.
5. Solve the two equations simultaneously to find the value of x. Rearrange the equations to solve for x:
x = 50 * cos(52), from the first equation
x = 50 * cos(43), from the second equation
We can use a calculator to evaluate the cosine function for both angles:
x ≈ 33.7609 km
x ≈ 38.7753 km
6. Determine the distance to the nearer warden. Since we are looking for the distance to the nearer warden, we choose the smaller value of x, which is approximately 33.7609 km.
So, the distance from C to the nearer warden is approximately 33.8 km (rounded to the nearest 0.1 km).