Question

An engineer is hired to design a bridge to carry traffic across the river. Her first problem is to determine the distance from point A to point B. To do so, she starts at point A and measures a distance of 250 m in a direction at a right angle to the segment AB. At point C, she uses her transit to find the angle ACB which measures 49°52'. How long should the bridge be?

Answer 1

To determine the length of the bridge, the engineer can use trigonometry by applying the tangent function to the measured baseline and angle, resulting in the bridge being approximately 184.21 meters long.

Step-by-step explanation:

The engineer wants to design a bridge and needs to measure the distance from point A to point B across a river. She measures a baseline distance of 250 m at a right angle from point A to point C and finds the angle ACB to be 49°52'. To find the length of the bridge, which is the distance from A to B, she can use trigonometry, particularly the tangent function (tan), because she has the opposite side (AC = 250 m) and needs to find the adjacent side (AB).

The tangent of angle ACB is equal to the opposite side divided by the adjacent side (tan(ACB) = AC / AB). Therefore, AB can be calculated as follows:

AB = AC / tan(ACB)

First, convert the angle to decimal degrees:

49°52' = 49 + (52/60) degrees
49°52' = 49.8667 degrees

Now, calculate AB:

AB = 250 m / tan(49.8667 degrees)

Using a calculator with trigonometric function capabilities:

AB = 250 m / tan(49.8667 degrees)

AB ≈ 184.21 m

Therefore, the length of the bridge should be approximately 184.21 meters.

Answer 2

To determine the length of the bridge, the engineer uses trigonometric functions to calculate the length of AB in the right triangle ABC, where AC is 250 m and angle ACB is 49°52'. The length calculated is approximately 213.75 m.

Step-by-step explanation:

The length of the bridge that the engineer needs to design can be found using trigonometry, specifically the tangent function. Given that point A and point B are on opposite sides of the river, and the engineer measures a distance of 250 m at a right angle from point A to point C, she forms a right triangle ABC. The angle at point C is given as 49°52' (which is equivalent to 49.867 degrees). To find the length of segment AB, which is the distance across the river, we use the tangent of the angle:
Tangent of angle ACB = Opposite / Adjacent
Therefore, AB = AC / Tan(ACB)
By substituting the known values: AB = 250 m / Tan(49.867 degrees), we can calculate the required length of the bridge.
Using a calculator, we find that Tan(49.867 degrees) is approximately 1.1696, which means AB (the length of the bridge) is approximately 213.75 m.