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Let's break down the stair design problem step by step:

### Given:
- The **rise** (vertical height per step) is 7 inches.
- The **run** (horizontal depth per step) can be between 10 and 12 inches.
- The total height of the stairs and floor space length is to be determined based on chosen values.

#### To solve:
1. **Length of floor space**: This is the total horizontal distance occupied by the stairs.
2. **Height of the stairs**: This is the total vertical distance (the "stair height") from the base to the next floor level.
3. **Number of stairs required**: This can be calculated using the total height and the rise per step.
4. **Optimum measure of the run**: This can be determined based on the most appropriate angle and step proportion, using the Pythagorean theorem.

### Step 1: Define some dimensions
Let's choose a **total height** for the stairs. For example:
- **Height of stairs**: Assume the height from the ground to the next floor is 84 inches (7 feet).

### Step 2: Calculate the number of stairs
Given that the rise per step is 7 inches, the number of steps required would be:
\[
\text{Number of stairs} = \frac{\text{Total height}}{\text{Rise per step}} = \frac{84}{7} = 12 \text{ steps}
\]

### Step 3: Length of floor space
Now, let's calculate the **length of the floor space**. The run per step can vary between 10 and 12 inches. We can calculate the total length of the floor space for both scenarios.

1. **Run = 10 inches**:
   \[
   \text{Total floor space} = \text{Run per step} \times \text{Number of steps} = 10 \times 12 = 120 \text{ inches}
   \]
   This is equivalent to 10 feet.

2. **Run = 12 inches**:
   \[
   \text{Total floor space} = 12 \times 12 = 144 \text{ inches} = 12 \text{ feet}
   \]

### Step 4: Use the Pythagorean theorem to find the diagonal length (stair length)
Using the Pythagorean theorem, the diagonal length (hypotenuse) of each stair can be calculated. For one stair step:
\[
\text{Diagonal (step length)} = \sqrt{\text{Rise}^2 + \text{Run}^2}
\]
For **Run = 10 inches**:
\[
\text{Step length} = \sqrt{7^2 + 10^2} = \sqrt{49 + 100} = \sqrt{149} \approx 12.2 \text{ inches}
\]
For **Run = 12 inches**:
\[
\text{Step length} = \sqrt{7^2 + 12^2} = \sqrt{49 + 144} = \sqrt{193} \approx 13.9 \text{ inches}
\]

### Conclusion:
- **Length of floor space**: Between 10 feet (for 10-inch run) and 12 feet (for 12-inch run).
- **Height of stairs**: 84 inches (7 feet).
- **Number of stairs**: 12 steps.
- **Optimum run**: Either 10 inches or 12 inches, depending on space availability and comfort (10 inches provides steeper stairs, 12 inches provides a more comfortable, shallower step).

Would you like to modify the values or go into further details?

### Related Questions:
1. How would the design change if the rise was 6 inches instead of 7?
2. What is the importance of using the Pythagorean theorem in staircase design?
3. How does the run affect the overall comfort of stair climbing?
4. How could you adjust the design if you had limited floor space?
5. What other architectural considerations should be taken into account when designing stairs?

**Tip:** Ensuring the correct balance between rise and run in stair design ensures both safety and comfort for users.

You own a company that designs and installs stairways for homes. A customer wants you to design and install a stairway for their home. Determine the total length of the floor space and the height of the stairs to reach the next floor level. The rise for each step is 7 inches and the run should be between 10 and 12 inches.

Learn how to design a stairway for homes by calculating the total floor space, the height of the stairs, and the number of steps using the Pythagorean theorem. This problem covers essential geometry and trigonometry concepts. Ideal for students in Grades 9-12.

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