A statue 1.6m tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60 degrees and from the same point the angle of elevation of the top of the pedestal is 45 degrees. Find the height of the pedestal.

To find the height of the pedestal, we can use trigonometry. Let's break down the problem step by step.

Understanding the Angles

We have two angles of elevation:

• The angle to the top of the statue is 60 degrees.
• The angle to the top of the pedestal is 45 degrees.

Setting Up the Problem

Let:

• h = height of the pedestal
• Height of the statue = 1.6 m
• Total height from the ground to the top of the statue = h + 1.6 m

Using Trigonometric Ratios

From the point on the ground, we can use the tangent function, which relates the angle of elevation to the opposite side (height) and the adjacent side (distance from the statue).

For the Top of the Statue

Using the angle of 60 degrees:

tan(60°) = (h + 1.6) / d

For the Top of the Pedestal

Using the angle of 45 degrees:

tan(45°) = h / d

Since tan(45°) = 1, we have:

h = d

Substituting Values

Now, substitute h = d into the equation for the statue:

tan(60°) = (d + 1.6) / d

Since tan(60°) = √3, we can rewrite the equation:

√3 = (d + 1.6) / d

Solving for d

Cross-multiplying gives:

√3 * d = d + 1.6

Rearranging leads to:

(√3 - 1) * d = 1.6

Thus:

d = 1.6 / (√3 - 1)

Finding the Height of the Pedestal

Now, substituting d back into h:

h = d = 1.6 / (√3 - 1)

Calculating this gives:

h ≈ 1.6 / 0.732 = 2.18 m (approximately)

Therefore, the height of the pedestal is approximately 2.18 meters.