To find the height of the pedestal, we use trigonometry, specifically the tangent function. Let’s denote the height of the pedestal as h and the total height (pedestal plus statue) as H.
Angle of Elevation to Top of Statue (60°): Using tan(60°) = √3, the equation is √3 = H/d, where d is the distance from the point to the pedestal.
Angle of Elevation to Top of Pedestal (45°): Using tan(45°) = 1, the equation is 1 = h/d. So, h = d.
From the first equation, H = d√3. Since H = h + 1.6 m (height of the pedestal plus the statue), substituting h gives d√3 = d + 1.6. Solving for d, we find d ≈ 0.93 m. Therefore, the height of the pedestal is approximately 0.93 meters.
Let’s discuss in detail
Trigonometric Application in Height Measurement
Trigonometry, a fundamental branch of mathematics, is widely used in various practical scenarios, including architecture and surveying. One common application is determining the height of structures when direct measurement is not feasible. The problem at hand involves using trigonometry to find the height of a pedestal on which a statue stands. By analyzing the angles of elevation from a specific point on the ground to the top of the statue and the top of the pedestal, we can calculate the pedestal’s height. This scenario exemplifies the practical utility of trigonometry in solving real-world problems.
Understanding the Problem: Statue on a Pedestal
The problem presents a statue that is 1.6 meters tall, standing on top of a pedestal. From a point on the ground, the angle of elevation to the top of the statue is 60°, and to the top of the pedestal is 45°. The objective is to determine the height of the pedestal. This setup forms two right-angled triangles – one from the point to the top of the statue and another from the point to the top of the pedestal.
The Role of Tangent in Angle of Elevation
In trigonometry, the tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. This function is particularly useful in scenarios where the height of an object needs to be determined from a specific distance. By applying the tangent function to the given angles of elevation, we can calculate the distances involved and, subsequently, the height of the pedestal.
Calculating the Distance to the Pedestal
First, we calculate the distance from the point to the pedestal using the 45° angle of elevation. The tangent of 45° is 1, so the equation is tan(45°) = (height of the pedestal)/d, where d is the distance. Since tan(45°) = 1, it follows that the height of the pedestal is equal to the distance d.
Determining the Total Height to the Statue’s Top
Next, we use the 60° angle of elevation to find the total height from the point to the top of the statue. Applying tan(60°) = √3, the equation is √3 = H/d, where H is the total height (pedestal plus statue). Since H = (height of the pedestal) + 1.6 meters, we substitute d for the height of the pedestal and solve for d.
Calculating the Pedestal’s Height
Solving the equation d√3 = d + 1.6 meters, we find the distance d and, consequently, the height of the pedestal. The height of the pedestal is approximately 0.93 meters. This calculation demonstrates the practical application of trigonometry in determining the height of structures and emphasizes the importance of mathematical principles in architectural and surveying tasks. Trigonometry proves to be an invaluable tool in scenarios where direct measurement is challenging, offering a reliable method for accurate estimation.