A building, B, is observed from two points P and Q which are 250 m apart. The angles PQB and QPB are found to be 67 degrees and 73 degrees respectively. What is the distance, to the nearest metre, of P from the building?
angle B is 180-67-73 = 40°
Now, using the law of sines
250/sin40° = PB/sin67°
Well, well, well! Looks like P and Q are having quite an observation party, huh? Let's join in on the fun, shall we?
Now, to find the distance of P from the building B, we can use some good old trigonometry. Let's call the distance we're looking for x meters.
Since we have the angles PQB and QPB, we can use the tangent function to determine the distance. The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side.
For angle PQB:
tan(67°) = x / 250
And for angle QPB:
tan(73°) = x / 250
Now, let's solve for x using these equations:
x = 250 * tan(67°) ≈ 632.72 metres
x = 250 * tan(73°) ≈ 783.26 metres
So, to the nearest meter, the distance of P from the building B is approximately 633 metres. Just be careful not to get too close to the edge, P!
To solve this problem, we can use the concept of trigonometry and the Law of Sines. Let's define the following information:
- Distance from P to the building B: x (the distance we need to find)
- Angle PQB: 67 degrees
- Angle QPB: 73 degrees
- Distance between P and Q: 250 m
We can begin by using the Law of Sines to establish a relationship between the side opposite to angle PQB, the side opposite to angle QPB, and their corresponding angles. The Law of Sines states:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we can write:
250 / sin(67) = x / sin(73)
Now we can solve this equation for x. Let's calculate the value:
x = (250 * sin(73)) / sin(67) ≈ 268.2 m
Therefore, the distance from P to the building B is approximately 268.2 meters.
To find the distance of point P from the building B, we can use the principles of trigonometry.
Let's draw a diagram to visualize the situation:
```
P
/|
/ |
67/ |B
/ |
/73 |
Q_____|
```
We are given the angles PQB = 67 degrees and QPB = 73 degrees, and the distance between points P and Q is 250 meters.
We can use the tangent function to find the height of the building. The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, we want to find the opposite side, which is the height of the building, and the adjacent side, which is the distance between P and B.
Let's define:
- h = height of the building (opposite side)
- d = distance between point P and building B (adjacent side)
Using the tangent function, we have:
tan(PQB) = h / d
tan(67) = h / d
Rearranging the formula to solve for h, we have:
h = d * tan(67)
Substituting the given value, we have:
h = 250 * tan(67)
Using a scientific calculator, we can find the value of tan(67) to be approximately 2.70.
Calculating the height, we have:
h = 250 * 2.70 = 675 meters
Therefore, the distance from point P to the building is approximately 675 meters.