Calculating Platform Height Using Angle of Elevation

This problem involves using trigonometry to find the height of a platform, given the height of a tower, the horizontal distance between the platform and the tower, and the angle of elevation from the top of the platform to the top of the tower.

Let's visualize the scenario. We have a platform and a tower standing vertically. The horizontal distance between their bases is given. An observer at the top of the platform looks up at the top of the tower. The angle formed by the line of sight and the horizontal line from the observer's eye level is the angle of elevation.

We can model this situation using a right-angled triangle. Consider a point P at the top of the platform and a point T at the top of the tower. Let B be the base of the platform and D be the base of the tower. The horizontal distance between the platform and the tower is the distance BD.

Draw a horizontal line from the top of the platform (P) meeting the tower at a point C. This forms a right-angled triangle PCT, where the angle of elevation is $\angle$TPC = 30°. The horizontal distance PC is equal to the horizontal distance BD, which is $40\sqrt{3}$ m. The vertical distance CT is the difference in height between the top of the tower and the top of the platform.

• Height of the tower (TD) = 45 m
• Horizontal distance (BD = PC) = $40\sqrt{3}$ m
• Angle of elevation ($\angle$TPC) = 30°
• Let the height of the platform (PB) be $h$ meters.

The height of the tower (TD) is equal to the height of the platform (PB) plus the height difference CT.

TD = PB + CT

45 = $h$ + CT

In the right-angled triangle PCT, we know the angle of elevation (30°) and the adjacent side (PC = $40\sqrt{3}$ m). We want to find the opposite side (CT). The trigonometric ratio that relates the opposite side and the adjacent side is the tangent function.

$\tan(\text{angle}) = \frac{\text{Opposite side}}{\text{Adjacent side}}$

Applying this to triangle PCT:

$\tan(30°) = \frac{\text{CT}}{\text{PC}}$

Substitute the known values:

$\tan(30°) = \frac{\text{CT}}{40\sqrt{3}}$

We know that the value of $\tan(30°)$ is $\frac{1}{\sqrt{3}}$.

So, $\frac{1}{\sqrt{3}} = \frac{\text{CT}}{40\sqrt{3}}$

To find CT, multiply both sides by $40\sqrt{3}$:

$\text{CT} = \frac{1}{\sqrt{3}} \times 40\sqrt{3}$

$\text{CT} = 40 \text{ m}$

Now we know the height difference CT is 40 m. We can use the relationship between the height of the tower, the height of the platform, and CT:

Height of tower = Height of platform + CT

45 m = $h$ + 40 m

To find the height of the platform ($h$), subtract 40 from 45:

$h = 45 - 40$

$h = 5 \text{ m}$

Therefore, the height of the platform is 5 m.

Revision Table: Key Steps in Height Calculation

Additional Information on Angle of Elevation and Trigonometry

The concept of angle of elevation is fundamental in height and distance problems in trigonometry. It is the angle measured upwards from the horizontal line to the line of sight to an object above the horizontal.

• Angle of Elevation: The angle formed by the horizontal line and the line of sight when looking upwards at an object.
• Angle of Depression: The angle formed by the horizontal line and the line of sight when looking downwards at an object.
• Trigonometric Ratios: In a right-angled triangle, the ratios of the sides are related to the angles. The three main ratios are sine (sin), cosine (cos), and tangent (tan).
  • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
  • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
  • $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

For standard angles like 30°, 45°, and 60°, the values of these trigonometric ratios are often used. For 30°:

• $\sin(30°) = \frac{1}{2}$
• $\cos(30°) = \frac{\sqrt{3}}{2}$
• $\tan(30°) = \frac{1}{\sqrt{3}}$

Understanding how to form a right-angled triangle from a given word problem and correctly applying the trigonometric ratios is key to solving these types of height and distance questions.