You have to design a staircase in a corridor. To minimize the "footprint" of the staircase, it has to be as steep as possible but still satisfy the following constraints.  The entrance should be 6.7' high. The horizontal width of each step should be at least 9.4". Individuals 6.3' tall should not have to bend their head so that it does not hit the ceiling. What should be the angle (in degrees) that the ceiling makes with the horizontal?

To solve this problem, we need to find the angle \(\theta\) that the ceiling makes with the horizontal to ensure that a person 6.3 feet tall does not have to bend their head when walking up the staircase.

### Step 1: Identify the given data
- The entrance height of the staircase: \( h_1 = 6.7 \) feet.
- The height of the person: \( h_2 = 6.3 \) feet.
- Minimum horizontal width of each step (tread): \( d = 9.4 \) inches (which is \( \frac{9.4}{12} = 0.7833 \) feet).

### Step 2: Determine the vertical clearance needed
For a person who is \( h_2 = 6.3 \) feet tall, they should not hit their head on the ceiling. Since the staircase starts at \( h_1 = 6.7 \) feet, the difference in height between the staircase entrance and the person is:
\[
h_1 - h_2 = 6.7 - 6.3 = 0.4 \text{ feet}
\]
This is the minimum vertical clearance required.

### Step 3: Set up the relationship between the vertical rise, horizontal run, and the angle \(\theta\)
The vertical rise per step is \( \Delta h \), and the horizontal run per step is \( d = 0.7833 \) feet. 

If \( n \) is the number of steps, the total horizontal distance is \( nd \), and the total vertical rise is \( n\Delta h \).

The key equation relating the vertical clearance and the staircase's footprint is:
\[
h_1 - n\Delta h = (nd) \tan(\theta) + 0.4
\]

This equation ensures that even at the farthest step, the vertical distance from the ceiling to the step below must be equal to or greater than the height of the person. Rearranging, we find:
\[
h_1 - n\Delta h = n \tan(\theta) d + 0.4
\]

### Step 4: Angle Calculation
Since each step's horizontal run is \( d \), and the angle is steepest when \( n\Delta h \) is minimized:
\[
\theta = \arctan\left(\frac{h_1 - 0.4}{d}\right)
\]

Substitute the values:
\[
\theta = \arctan\left(\frac{6.7 - 0.4}{0.7833}\right)
\]

Let's calculate this:

\[
\theta = \arctan\left(\frac{6.3}{0.7833}\right) \approx \arctan(8.04)
\]

Using the arctan function, we find:
\[
\theta \approx 82.89^\circ
\]

### Final Answer
The angle that the ceiling should make with the horizontal is approximately **82.89 degrees**.

Would you like further details or have any other questions?

### Further Questions:
1. How does altering the minimum tread width affect the angle calculation?
2. What would happen if the staircase needed to accommodate a person taller than 6.3 feet?
3. How would increasing the entrance height to 7 feet change the angle?
4. What are the practical implications of a staircase this steep?
5. How does the calculation change if the minimum headroom clearance needs to be larger?

### Tip:
Always double-check the unit conversions, especially when dealing with measurements in both inches and feet, to avoid calculation errors.

Learn how to calculate the ceiling angle for a steep staircase design to ensure sufficient headroom clearance. This problem involves trigonometric calculations using the arctangent function and considerations in geometric design. Suitable for architects and engineers.

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