From A, due east of a building, the angle of elevation of the top is 45 degree. From B, southwest of the building , the angle of elevation is 30 degree. If AB=250m, find the heigt of the building.

let C be the third vertex of the horizontal triangle, directly below the building

for the vertical triangles, height = AC(tan(45)) = BC(tan(30)) = AC, since tan(45) = 1

now use law of cosines on the horizontal triangle to obtain

250^2 = AC^2 + (AC/tan(30))^2 – 2(AC)(AC/tan(30))cos(135)

and solve the quadratic equation to obtain AC = height = 98.4413

To find the height of the building, we can use trigonometry. Let's break down the problem and solve it step by step.

Step 1: Draw a diagram
Draw a diagram to visualize the situation. Label points A, B, and the building as shown below:

A (due east) B (southwest)
----------------- Building -----------------

Step 2: Identify relevant angles and distances
From point A, the angle of elevation of the top of the building is given as 45 degrees. From point B, the angle of elevation is given as 30 degrees. The distance between points A and B (AB) is given as 250m.

Step 3: Break down the problem into trigonometric ratios
Using trigonometry, we can relate the angle of elevation to the height of the building. The tangent ratio is defined as:

tangent(angle) = opposite/adjacent

In our case, the opposite side is the height of the building, and the adjacent side is the horizontal distance from each point to the building.

Step 4: Calculate the height of the building from point A
From point A, the angle of elevation is 45 degrees. Let's set up the equation using the tangent ratio:

tan(45 degrees) = height of the building / distance from A to the building

Simplifying the equation:

1 = height of the building / distance from A to the building

Since the distance from A to the building is not given, we need to find it using the given information. We know that AB = 250m, so we need to divide it by the tangent of the angle between AB and the line due east.

Step 5: Find the distance from A to the building
The angle between AB and the line due east is 45 degrees (complementary to the angle of elevation from point A). We can find the distance from A to the building as follows:

distance from A to the building = AB / tan(45 degrees)

Calculating the value:

distance from A to the building = 250m / tan(45 degrees)

distance from A to the building ≈ 250m

Step 6: Substitute the values into the equation
Now that we know the distance from A to the building, we can substitute the values into the equation we set up earlier:

1 = height of the building / distance from A to the building

1 = height of the building / 250m

Simplifying the equation:

height of the building = 250m

Therefore, the height of the building from point A is approximately 250m.

Step 7: Calculate the height of the building from point B
Using a similar process, we can find the height of the building from point B. The equation will be:

tan(30 degrees) = height of the building / distance from B to the building

Since the width of the building is not given, we need to find it. Since point B is southwest of the building, we can use the distance formula to calculate the distance from B to the building as follows:

distance from B to the building = sqrt((AB^2) + (AB^2))

Calculating the value:

distance from B to the building = sqrt((250m ^ 2) + (250m ^ 2))

distance from B to the building ≈ 353.55m

Substituting the values into the equation:

tan(30 degrees) = height of the building / distance from B to the building

tan(30 degrees) = height of the building / 353.55m

Simplifying the equation:

height of the building ≈ tan(30 degrees) * 353.55m

Calculating the value:

height of the building ≈ 0.577 * 353.55m

height of the building ≈ 203.71m

Therefore, the height of the building from point B is approximately 203.71m.

Note: The height calculated from point A and point B may differ due to the different angles of elevation.