From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30∘ and 45∘, respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.
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Step-by-Step Solution
Step 1: Visualize the Problem and Define Variables
We first visualize the scenario. Let P be the point on the bridge where the observation is made. Let D be the point on the river bank directly below P. Let A and B be points on the opposite banks. The problem states that the height of the bridge from the banks is 3 m, so PD=3 m. This forms two right-angled triangles, △PDA and △PDB.
Step 2: Identify Angles of Depression and Alternate Interior Angles
The angles of depression are given from point P to the points A and B on the banks. Since the horizontal line through P is parallel to the river banks, the angle of depression to point A is equal to the alternate interior angle ∠PAD=30∘. Similarly, the angle of depression to point B is equal to the alternate interior angle ∠PBD=45∘.
Step 3: Calculate Distance AD Using Trigonometry in △PDA
In the right-angled triangle △PDA, we know the opposite side PD=3 m and the angle ∠PAD=30∘. We want to find the adjacent side AD. The tangent function relates these quantities: tan(θ)=adjacentopposite. Substituting the known values, we get tan(30∘)=AD3.
Step 4: Solve for AD
From the previous step, we have AD=tan(30∘)3. We know that tan(30∘)=31. Substituting this value, we find that AD=33 m.
Step 5: Calculate Distance BD Using Trigonometry in △PDB
Now, consider the right-angled triangle △PDB. We know the opposite side PD=3 m and the angle ∠PBD=45∘. We want to find the adjacent side BD. Using the tangent function again, we have tan(45∘)=BD3.
Step 6: Solve for BD
From the equation BD=tan(45∘)3, and knowing that tan(45∘)=1, we can calculate BD. Thus, BD=3 m.
Step 7: Calculate the Width of the River
The width of the river is the sum of the distances AD and BD, since D lies between A and B. Adding the values we calculated for AD and BD, we get the total width AB=33+3 m. This can be factored as 3(3+1) m.