To find the Sun's altitude at the given instant, we can use the concept of trigonometry. The tangent of the Sun's altitude is given by the ratio of the height of the tower to the length of its shadow. Let's calculate this step-by-step:

Step 1: Write the tangent relation

$\tan(\theta) = \frac{\text{Height of the tower}}{\text{Length of the shadow}}$

Where:

• Height of the tower = $20\sqrt{3}$ m
• Length of the shadow = $60$ m

$\tan(\theta) = \frac{20\sqrt{3}}{60}$

Step 2: Simplify the ratio

$\tan(\theta) = \frac{\sqrt{3}}{3}$

Step 3: Find $\theta$ (Sun's altitude)

The angle $\theta$ is the angle whose tangent is $\frac{\sqrt{3}}{3}$. From trigonometric tables or unit circle values, we know:

$\tan(30^\circ) = \frac{\sqrt{3}}{3}$

Thus, the Sun's altitude, $\theta$, is $30^\circ$.

Final Answer:

The Sun’s altitude at that instant is $30^\circ$.

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Do you want more details or have any questions?
Here are 5 related questions for further practice:

1. What is the tangent of a $45^\circ$ angle, and how can it be used in shadow problems?
2. If the shadow length changes to 50 m, what would the Sun's altitude be for the same tower?
3. Derive the relationship between shadow length and Sun’s altitude for a given height.
4. How does the height of an object affect the angle of elevation for a fixed shadow length?
5. Solve a similar problem with a tower height of 30 m and a shadow length of 90 m.

Tip: For trigonometric problems, always check the standard values for sine, cosine, and tangent to simplify your calculations quickly.