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The image shows a roof structure where the rise and run are indicated, and the table below is asking for the height (h) to be filled in based on the slope ratio provided as \(3:12\), along with the total run of \(40 \, \text{ft} \, 4 \, \text{in.}\).

Here's how we can calculate the height (h) based on the slope and run:

### Step-by-step approach:
1. **Slope Ratio**: \(3:12\) means that for every 12 feet of horizontal run, the rise is 3 feet.
   
2. **Run**: The total run is given as \(40 \, \text{ft} \, 4 \, \text{in.}\).
   - Convert \(40 \, \text{ft} \, 4 \, \text{in.}\) to feet:
     \[
     40 \, \text{ft} + \frac{4 \, \text{in.}}{12} = 40 + 0.3333 = 40.3333 \, \text{ft}.
     \]

3. **Proportional Rise**: Using the ratio \(3:12\), we calculate the rise for a run of \(40.3333 \, \text{ft}\). Since the rise is 3 for every 12 feet of run:
   \[
   \text{Rise} = \frac{3}{12} \times 40.3333 = 10.0833 \, \text{ft}.
   \]

4. **Convert Rise to Feet and Inches**:
   - The height \(h\) is \(10.0833 \, \text{ft}\).
   - The decimal part \(0.0833 \, \text{ft}\) can be converted to inches:
     \[
     0.0833 \times 12 = 1 \, \text{in}.
     \]
   - Therefore, the height \(h\) is approximately **10 ft 1 in**.

### Final Answer:
The height (h) is \(10 \, \text{ft} \, 1 \, \text{in.}\). 

Would you like more details or have any questions? Here are some related topics:

1. How do you calculate the slope of a roof?
2. What is the significance of the rise-to-run ratio?
3. How can you convert between feet and inches?
4. What is the formula for calculating height using slope and run?
5. How do you calculate roof pitch angles from rise-to-run ratios?

**Tip**: When converting inches to feet, remember that 1 foot equals 12 inches.

What is the height (h) of the roof if the total run is 40 ft 4 in, and the slope ratio is 3:12?

Learn how to calculate the height (h) of a roof with a slope ratio of 3:12 and a total run of 40 ft 4 in. Step-by-step solution, including unit conversions and proportional rise calculation.

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