From a point P on the ground the angle of elevation of the top of a 10 m tall building is $30^\circ$. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is $45^\circ$. Find the length of the flagstaff and the distance of the building from the point P. (You may take $\sqrt{3} = 1.732$)

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 $\triangle PAB$. The angle of elevation of the top of the building (A) from point P is $30^\circ$. 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 $\tan 30^\circ = \frac{1}{\sqrt{3}}$ 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 $\triangle PAC$. 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 $45^\circ$. We again use the tangent function.

Step 5: Calculate the total height AC

We know that $\tan 45^\circ = 1$ and we have calculated $AP = 10\sqrt{3}$ 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 $\sqrt{3}$ and calculate numerical values

Finally, we substitute the approximate value of $\sqrt{3} = 1.732$ 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.

More from this chapter

• **निम्नलिखित दो (02) प्रश्नों के लिए:** एक इमारत के शीर्ष से h ऊँचाई वाले एक टावर के शीर्ष और तल के अवनमन कोण क्रमशः α और β हैं। मान लीजिए कि इमारत की ऊँचाई H ह
• A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of 30 degrees with it. The distance betwee
• From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60 degrees and the angle of depression of its foot is 45 degrees. Det
• The angle of elevation of the top of a building from a point on the ground is 30 degrees. On walking 20 m towards the building the angle becomes 60 degrees. Fin
• A contractor plans to install a slide for children under 5. The top of the slide is at a height of 1.5 m, and it is inclined at an angle of 30° to the ground. W
• A 1.5m tall boy is standing at some distance from a 32m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60°
• The angle of elevation of a cloud from a point h meters above a lake is α and the angle of depression of its reflection in the lake is β. Prove that the height
• A vertical pole 12 m high casts a shadow 5√3 m long. Find the angle of elevation of the Sun.

All Some Applications of Trigonometry solutions →