From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30∘. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is 45∘. Find the length of the flagstaff and the distance of the building from the point P. (You may take 3=1.732)
Get the complete, step-by-step math solution for: "From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30^ . A flag is hoisted at the top of the building and the a...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Define the setup and variables
We first define the various components of the problem. AB represents the height of the building, BC is the length of the flagstaff. P is the point on the ground from where the observations are made, and AP is the distance from the point P to the base of the building.
Step 2: Analyze the first triangle (PAB)
Consider the right-angled triangle △PAB. The angle of elevation of the top of the building (A) from point P is 30∘. We use the tangent function, which relates the opposite side (AB, height of the building) to the adjacent side (AP, distance from the building).
Step 3: Calculate the distance AP
We know that tan30∘=31 and the height of the building AB=10 m. Substituting these values into our equation, we can solve for AP. This gives us the distance of the building from point P.
Step 4: Analyze the second triangle (PAC)
Now consider the larger right-angled triangle △PAC. Here, AC is the total height of the building and the flagstaff (AC=AB+BC). The angle of elevation of the top of the flagstaff (C) from P is 45∘. We again use the tangent function.
Step 5: Calculate the total height AC
We know that tan45∘=1 and we have calculated AP=103 m. Substituting these values into the equation from the previous step, we can find the total height AC.
Step 6: Calculate the length of the flagstaff BC
The length of the flagstaff BC is the difference between the total height AC and the height of the building AB. Substitute the calculated values for AC and the given value for AB to find the length of the flagstaff.
Step 7: Substitute the value of 3 and calculate numerical values
Finally, we substitute the approximate value of 3=1.732 into our expressions for AP and BC to get the numerical answers for the distance of the building from P and the length of the flagstaff.