Question

The angle of elevation of the top P of a tower from the feet of one person standing due South of the tower is 45° and from the feet of another person standing due west of the tower is 30°. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to

• A. \(\frac{5}{2}\sqrt5\)
• B. 10
• C. 5
• D. \(5\sqrt5\)

Answer 1

The Correct Option is B

Given:

• Height of the tower \( AB = 5 \, \text{m} \).
• Angles: \( \angle APB = 45^\circ \) and \( \angle PAB = 90^\circ \). 

Step 1: Use Trigonometry in \( \triangle ABP \)

In \( \triangle ABP \), we know:

\[ \tan 45^\circ = \frac{AB}{AP}. \]

Since \( \tan 45^\circ = 1 \):

\[ 1 = \frac{5}{AP}. \]

Rearranging, we get:

\[ AP = 5 \, \text{m}. \]

Step 2: Use Trigonometry in \( \triangle ABQ \)

In \( \triangle ABQ \), \( \angle BAQ = 30^\circ \). Using the tangent formula:

\[ \tan 30^\circ = \frac{AB}{AQ}. \]

Substitute \( \tan 30^\circ = \frac{1}{\sqrt{3}} \):

\[ \frac{1}{\sqrt{3}} = \frac{5}{AQ}. \]

Rearranging, we find:

\[ AQ = 5\sqrt{3} \, \text{m}. \]

Step 3: Calculate the Hypotenuse \( PQ \)

Using the Pythagoras theorem in \( \triangle APQ \):

\[ PQ^2 = AP^2 + AQ^2. \]

Substitute \( AP = 5 \, \text{m} \) and \( AQ = 5\sqrt{3} \, \text{m} \):

\[ PQ^2 = 5^2 + (5\sqrt{3})^2. \]

Simplify:

\[ PQ^2 = 25 + 75 = 100. \]

Therefore:

\[ PQ = \sqrt{100} = 10 \, \text{m}. \]

Final Answer:

The length of \( PQ \) is \( 10 \, \text{m} \).