Sam is framing a closet under a stairway. The stairway is 16 ft 9 in.​ long, and its angle of elevation is 39degrees. Find the depth of the closet to the nearest inch.

To find the depth of the closet, we can treat this as a right triangle problem where the stairway acts as the hypotenuse, the depth of the closet is the adjacent side, and the angle of elevation is given. Using trigonometry, we can apply the cosine function since it relates the adjacent side (depth of the closet) to the hypotenuse (length of the stairway).

### Step-by-Step Solution:

1. **Convert the length of the stairway to feet:**
   - \( 16 \, \text{ft} \, 9 \, \text{in} \) means 16 feet + 9 inches.
   - Convert 9 inches to feet: \( \frac{9}{12} = 0.75 \, \text{ft} \).
   - So, the total length of the stairway is \( 16 + 0.75 = 16.75 \, \text{ft} \).

2. **Use the cosine function:**
   The cosine of the angle is given by:
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
   \]
   Where:
   - \( \theta = 39^\circ \) (angle of elevation),
   - Hypotenuse \( = 16.75 \, \text{ft} \),
   - Adjacent (depth of the closet) = \( d \).

   Therefore, 
   \[
   \cos(39^\circ) = \frac{d}{16.75}
   \]

3. **Solve for the depth \( d \):**
   \[
   d = 16.75 \times \cos(39^\circ)
   \]
   Now, calculate this value:

\[
d \approx 16.75 \times 0.777 = 13.02 \, \text{ft}
\]

4. **Convert the depth to inches:**
   \[
   13.02 \, \text{ft} \times 12 \, \text{in/ft} \approx 156.24 \, \text{inches}
   \]
   To the nearest inch, the depth of the closet is approximately **156 inches**.

### Final Answer:
The depth of the closet is approximately **156 inches**.

Would you like more details or have any further questions?

Here are some related questions to expand this topic:
1. How can the sine function be used to find the height of the stairway above the floor?
2. If the angle of elevation changed, how would the depth of the closet be affected?
3. How would the depth change if the stairway were longer but the angle stayed the same?
4. Can the depth be calculated using the tangent function instead of cosine? Why or why not?
5. How would the calculation change if the length of the stairway was measured in meters instead of feet?

**Tip:** When working with angles and distances, converting between units (feet, inches, meters) before applying trigonometric functions helps avoid rounding errors later in the calculations.

Sam is framing a closet under a stairway. The stairway is 16 ft 9 in. long, and its angle of elevation is 39 degrees. Find the depth of the closet to the nearest inch.

Discover how to find the depth of a closet framed under a stairway using trigonometry. The problem involves calculating the depth given the stairway's length and angle of elevation. Learn to use the cosine function to solve real-world right triangle problems, suitable for students in Grades 9-12.

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