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We are given that:

- The height of the antenna is 7 feet.
- The angle of elevation of the sun is 63 degrees.
- The angle at the top of the house is 140 degrees.

To find the length of the shadow, we can use trigonometry. The problem involves a right triangle, where:
- The height of the antenna is the opposite side,
- The length of the shadow is the adjacent side, 
- The angle of elevation (63 degrees) is the angle between the shadow and the line of the sun's rays.

We can use the tangent function to solve for the length of the shadow:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Substituting the values:

\[
\tan(63^\circ) = \frac{7}{\text{shadow}}
\]

Now, let's solve for the length of the shadow.The length of the shadow that the antenna casts on the roof of the house is 3.6 feet.

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How does the length of the shadow change if the sun's elevation increases to 70 degrees?
2. What happens to the shadow length if the antenna is raised to 10 feet?
3. How would the calculation change if the roof had a different angle at the peak?
4. How can this problem be applied to calculate shadows for much taller structures, like towers?
5. How would you calculate the total shadow length if the house also casts a shadow?

**Tip:** When using trigonometric functions, ensure that your calculator or programming environment is in the correct angle mode (degrees or radians).

A 7-foot antenna is placed on top of the peak of a house. The angle at the top of the house is 140 degrees. The angle of elevation of the sun is 63 degrees. What is the length of the shadow the antenna casts on the roof of the house?

Learn how to calculate the shadow length of a 7-foot antenna on a house roof using trigonometry. This problem explores the tangent function in a right triangle with an angle of elevation of 63 degrees and involves applying trigonometric formulas. Suitable for students in Grades 9-12.

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