A staircase is 30.4 -feet high and 77.5 -feet at the base. Determine the angle from its base to the top. Round to the nearest tenth of a degree. 

To determine the angle from the base of the staircase to the top, we can use trigonometry, specifically the tangent function. The tangent of the angle \(\theta\) can be found using the formula:

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

Here, the "opposite" side is the height of the staircase (30.4 feet), and the "adjacent" side is the base of the staircase (77.5 feet). So, the equation becomes:

\[
\tan(\theta) = \frac{30.4}{77.5}
\]

Now, we solve for \(\theta\) by taking the arctangent (inverse tangent) of both sides:

\[
\theta = \arctan\left(\frac{30.4}{77.5}\right)
\]

Let's calculate this value.The angle from the base of the staircase to the top is approximately \( 21.4^\circ \) when rounded to the nearest tenth of a degree.

Would you like more details on this calculation or have any other questions?

Here are some related questions you might find interesting:
1. How would the angle change if the height of the staircase were doubled?
2. What is the length of the staircase (hypotenuse) given the current dimensions?
3. How does the angle affect the steepness and safety of the staircase?
4. What is the importance of this angle in architecture and construction?
5. How would the base length need to change to achieve a 30-degree angle?

**Tip:** Knowing how to use trigonometric functions can help solve many real-world problems involving right triangles!

A staircase is 30.4 feet high and 77.5 feet at the base. Determine the angle from its base to the top. Round to the nearest tenth of a degree.

Learn how to determine the angle of a staircase that is 30.4 feet high and 77.5 feet at the base using trigonometry. This problem illustrates the use of the tangent function and is suitable for students in Grades 9-11.

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