Question 1179678: Two industrious Avenues students are trying to find the distance across the Pinheiros River in São
Paulo. After marking points A and B about 86 m apart on one bank, they sight a point P, on the opposite
bank, and measure angles PAB and PBA to be 30 and 129 degrees, respectively. This enables them to
calculate the altitude from P to the baseline AB. To the nearest meter, how wide, straight across, is the
Pinheiros River?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem using trigonometry:
1. Visualize the Triangle:
Imagine a triangle PAB, where A and B are points on one bank of the river, and P is a point on the opposite bank.
We know AB = 86 meters.
We know angle PAB = 30 degrees.
We know angle PBA = 129 degrees.
2. Calculate Angle APB:
The sum of angles in a triangle is 180 degrees.
Angle APB = 180 - (30 + 129) = 180 - 159 = 21 degrees.
3. Use the Law of Sines to Find AP:
The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C)
In our case: AP/sin(PBA) = AB/sin(APB)
AP/sin(129) = 86/sin(21)
AP = (86 * sin(129)) / sin(21)
AP ≈ (86 * 0.7771) / 0.3584
AP ≈ 234.8 meters
4. Calculate the Altitude (Width of the River):
Let h be the altitude from P to AB.
In triangle AP, we have: sin(PAB) = h/AP
sin(30) = h/234.8
h = 234.8 * sin(30)
h = 234.8 * 0.5
h = 117.4 meters
5. Round to the Nearest Meter:
The width of the river is approximately 117 meters.
Final Answer:
The width of the Pinheiros River is approximately 117 meters.
|
|
|
