Understanding the Problem: Tower and Building Height

This problem involves trigonometry, specifically using angles of elevation to find the height of a tower. We are given the height of a building and the angles of elevation to the top of the tower from two different points: the top of the building and the foot of the building.

Let's define the key elements:

• Height of the building = 680 m
• Angle of elevation from the top of the building to the top of the tower = 45°
• Angle of elevation from the foot of the building to the top of the tower = 60°
• We need to find the height of the tower.

Setting up the Geometry

Imagine a vertical tower and a vertical building some distance apart on level ground. Let:

• \(H\) be the height of the tower.
• \(x\) be the horizontal distance between the foot of the building and the foot of the tower.

We can visualize two right-angled triangles formed by these elements and the lines of sight corresponding to the angles of elevation.

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.

From the Foot of the Building:

The angle of elevation from the foot of the building to the top of the tower is 60°. The opposite side is the height of the tower (\(H\)), and the adjacent side is the horizontal distance between the building and the tower (\(x\)).

So, we have:

\(\tan(60^\circ) = \frac{\text{Height of Tower}}{\text{Distance between Building and Tower}}\)

\(\tan(60^\circ) = \frac{H}{x}\)

We know that \(\tan(60^\circ) = \sqrt{3}\). Therefore:

\(\sqrt{3} = \frac{H}{x}\)

This gives us our first equation:

Equation 1: \(H = x\sqrt{3}\)

From the Top of the Building:

The height of the building is 680 m. The angle of elevation from the top of the building to the top of the tower is 45°. The line of sight from the top of the building to the top of the tower is parallel to the ground. The vertical distance from the level of the top of the building up to the top of the tower is \(H - 680\) m. The horizontal distance is still \(x\).

So, we have:

\(\tan(45^\circ) = \frac{\text{Vertical distance from top of building to top of tower}}{\text{Distance between Building and Tower}}\)

\(\tan(45^\circ) = \frac{H - 680}{x}\)

We know that \(\tan(45^\circ) = 1\). Therefore:

\(1 = \frac{H - 680}{x}\)

This gives us our second equation:

Equation 2: \(x = H - 680\)

Solving for the Height of the Tower

Now we have a system of two linear equations with two variables, \(H\) and \(x\). We can substitute the expression for \(x\) from Equation 2 into Equation 1.

Substitute \(x = H - 680\) into \(H = x\sqrt{3}\):

\(H = (H - 680)\sqrt{3}\)

Distribute \(\sqrt{3}\) on the right side:

\(H = H\sqrt{3} - 680\sqrt{3}\)

Now, we need to isolate \(H\). Move the term with \(H\) from the right side to the left side:

\(H - H\sqrt{3} = -680\sqrt{3}\)

Factor out \(H\) from the terms on the left side:

\(H(1 - \sqrt{3}) = -680\sqrt{3}\)

To get a positive coefficient for \(H\), multiply both sides by -1:

\(H(\sqrt{3} - 1) = 680\sqrt{3}\)

Now, divide by \((\sqrt{3} - 1)\) to solve for \(H\):

\(H = \frac{680\sqrt{3}}{\sqrt{3} - 1}\)

To simplify and rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator, which is \((\sqrt{3} + 1)\):

\(H = \frac{680\sqrt{3}}{(\sqrt{3} - 1)} \times \frac{(\sqrt{3} + 1)}{(\sqrt{3} + 1)}\)

Multiply the numerators:

\(680\sqrt{3}(\sqrt{3} + 1) = 680(\sqrt{3} \times \sqrt{3} + \sqrt{3} \times 1) = 680(3 + \sqrt{3})\)

Multiply the denominators using the difference of squares formula \((a-b)(a+b) = a^2 - b^2\):

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

So, the expression for \(H\) becomes:

\(H = \frac{680(3 + \sqrt{3})}{2}\)

Finally, divide 680 by 2:

\(H = 340(3 + \sqrt{3})\)

The height of the tower is \(340(3 + \sqrt{3})\) meters.

Conclusion

Based on the angles of elevation and the height of the building, the height of the tower is calculated to be \(340(3 + \sqrt{3})\) m.

Revision Table: Key Concepts

Additional Information: Angles in Trigonometry

Understanding angles of elevation and depression is crucial in solving problems related to heights and distances.

• Angle of Elevation: When you look *up* at an object, the angle between your horizontal line of sight and your line of sight to the object is the angle of elevation.
• Angle of Depression: When you look *down* at an object, the angle between your horizontal line of sight and your line of sight to the object is the angle of depression. The angle of depression from point A to point B is equal to the angle of elevation from point B to point A, assuming the horizontal line is the same level.

Commonly used trigonometric values in these types of problems include:

These standard values help in quickly solving trigonometric equations encountered in height and distance problems.