Calculating Tower Height Using Angles of Elevation

This problem involves using trigonometry, specifically the tangent function, to find the height of a tower given the angles of elevation to the bottom and top of a flagstaff mounted on it.

Let's define the scenario:

• Let \(h\) be the height of the tower.
• Let \(f\) be the length of the flagstaff, given as 3 m.
• Let \(x\) be the horizontal distance from the observation point on the field to the base of the tower.

We have two angles of elevation from the same point on the field:

• Angle of elevation to the bottom of the flagstaff (top of the tower) is 30°.
• Angle of elevation to the tip of the flagstaff is 45°.

We can set up two right-angled triangles based on this information. The vertical sides of these triangles are the height to the bottom of the flagstaff (which is the tower height \(h\)) and the height to the tip of the flagstaff (which is the tower height plus the flagstaff length, \(h+f\)). The horizontal side for both triangles is the distance \(x\).

Applying Trigonometry (Tangent Function)

The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side.

For the angle of elevation to the bottom of the flagstaff (30°):

\( \tan(30^\circ) = \frac{\text{Height of the tower}}{\text{Distance from the point}} = \frac{h}{x} \)

For the angle of elevation to the tip of the flagstaff (45°):

\( \tan(45^\circ) = \frac{\text{Height of the tower + Height of flagstaff}}{\text{Distance from the point}} = \frac{h+f}{x} \)

Solving for the Height of the Tower

We know the values of \( \tan(30^\circ) \) and \( \tan(45^\circ) \):

• \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)
• \( \tan(45^\circ) = 1 \)

Substitute these values into our equations:

From the first equation:

\( \frac{1}{\sqrt{3}} = \frac{h}{x} \)

This gives us \( x = h\sqrt{3} \) (Equation 1)

From the second equation:

\( 1 = \frac{h+f}{x} \)

This gives us \( x = h+f \) (Equation 2)

We are given that \( f = 3 \) m. Substitute this into Equation 2:

\( x = h+3 \) (Equation 3)

Now we have two expressions for \(x\). We can set Equation 1 and Equation 3 equal to each other:

\( h\sqrt{3} = h+3 \)

Now, we need to solve for \(h\). Rearrange the equation to group terms with \(h\):

\( h\sqrt{3} - h = 3 \)

Factor out \(h\) from the left side:

\( h(\sqrt{3} - 1) = 3 \)

Divide by \( (\sqrt{3} - 1) \) to find \(h\):

\( h = \frac{3}{\sqrt{3} - 1} \)

To simplify and get a numerical value, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of \( (\sqrt{3} - 1) \), which is \( (\sqrt{3} + 1) \):

\( h = \frac{3}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} \)

\( h = \frac{3(\sqrt{3} + 1)}{(\sqrt{3})^2 - (1)^2} \)

\( h = \frac{3(\sqrt{3} + 1)}{3 - 1} \)

\( h = \frac{3(\sqrt{3} + 1)}{2} \)

Now, substitute the approximate value of \( \sqrt{3} \approx 1.732 \):

\( h \approx \frac{3(1.732 + 1)}{2} \)

\( h \approx \frac{3(2.732)}{2} \)

\( h \approx \frac{8.196}{2} \)

\( h \approx 4.098 \text{ m} \)

Comparing with Given Options

The calculated height of the tower is approximately 4.098 m. Let's look at the options provided:

1. 3.90 m
2. 4.00 m
3. 4.10 m
4. 4.25 m

Comparing 4.098 m with the options, 4.10 m is the closest value and provides the best approximation to the height of the tower.

The smallest difference is 0.002 m, which corresponds to the option 4.10 m.

Revision Table: Key Trigonometry Concepts for Height and Distance

Additional Information: Applications of Height and Distance Problems

Problems involving height and distance are common applications of trigonometry. They are used in various fields:

• Surveying: To determine distances and elevations of land points.
• Navigation: In celestial navigation or determining distances at sea.
• Engineering: For designing structures, roads, and bridges.
• Astronomy: To calculate distances between celestial bodies.
• Physics: In vector analysis and projectile motion.

These problems often involve setting up right-angled triangles and using sine, cosine, or tangent based on the given angles and sides.