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A statue 3 m 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°, and from the same point, the angle of elevation of the top of the pedestal is 30°. Find the height of the pedestal and its distance from the point of observation on the ground. (Use [tex]\sqrt{3} = 1.73[/tex])

Answer :

To solve this problem, we'll use trigonometry, which involves calculating distances and heights using angles. The problem gives us two angles of elevation and the height of the statue.

Let's break down the problem:


  1. Understand the scenario:


    • We have a statue that is 3 meters tall on top of a pedestal.

    • There are two angles of elevation observed from a point on the ground: 60° to the top of the statue and 30° to the top of the pedestal.



  2. Define the variables:


    • Let [tex]h_p[/tex] be the height of the pedestal.

    • Let [tex]d[/tex] be the distance from the point of observation to the base of the pedestal.



  3. Use trigonometric ratios:


    • For the angle of elevation to the top of the statue (60°), the total height from the top of the pedestal to the top of the statue is [tex]h_p + 3[/tex]. The tangent of the 60° angle relates the total height to the distance [tex]d[/tex]:
      [tex]\tan(60°) = \frac{h_p + 3}{d}[/tex]

    • For the angle of elevation to the top of the pedestal (30°), we have:
      [tex]\tan(30°) = \frac{h_p}{d}[/tex]



  4. Calculate using the trigonometric values:


    • [tex]\tan(60°) = \sqrt{3} = 1.73[/tex]

    • [tex]\tan(30°) = \frac{1}{\sqrt{3}} = \frac{1}{1.73}[/tex]



  5. Solve the equations:


    • From the equation [tex]\tan(30°) = \frac{h_p}{d}[/tex], we can express [tex]d[/tex] in terms of [tex]h_p[/tex]:
      [tex]d = \frac{h_p}{\tan(30°)} = \frac{h_p}{\frac{1}{1.73}} = 1.73h_p[/tex]

    • Substitute this into the first equation:
      [tex]1.73 = \frac{h_p + 3}{1.73h_p}[/tex]

    • Simplify and solve for [tex]h_p[/tex]:
      [tex]1.73^2 h_p = h_p + 3[/tex]

      • [tex]2.9929h_p = h_p + 3[/tex]

      • [tex]1.9929h_p = 3[/tex]

      • [tex]h_p = \frac{3}{1.9929} \approx 1.51 \, \text{m}[/tex]





  6. Calculate the distance [tex]d[/tex]:


    • [tex]d = 1.73 \times 1.51 \approx 2.61 \, \text{m}[/tex]




Therefore, the height of the pedestal is approximately 1.51 meters, and the distance from the point of observation to the base of the pedestal is approximately 2.61 meters.

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Rewritten by : Barada