The problem involves finding the length of a ladder that leans against the side of a house, given:

1. The angle of elevation of the ladder: $62^\circ$.
2. The distance from the bottom of the ladder to the side of the house: $12 \, \text{ft}$.

The situation forms a right triangle, where:

• The ladder is the hypotenuse ($L$).
• The distance from the bottom of the ladder to the house is the adjacent side ($12 \, \text{ft}$).
• The angle of elevation is $62^\circ$.

Formula:

Using the cosine function: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ $\cos(62^\circ) = \frac{12}{L}$

Rearranging to solve for $L$ (the ladder's length): $L = \frac{12}{\cos(62^\circ)}$

Calculation:

1. Find $\cos(62^\circ)$ using a calculator: $\cos(62^\circ) \approx 0.4695$
2. Substitute into the equation: $L = \frac{12}{0.4695} \approx 25.56 \, \text{ft}$

Final Answer:

The length of the ladder is approximately 25.6 ft (rounded to the nearest tenth).

Would you like a detailed step-by-step explanation or have any questions?

Here are 5 related questions to expand your understanding:

1. How do we calculate the height of the house using the sine function in this scenario?
2. If the ladder's angle of elevation changed to $45^\circ$, how would the ladder length differ?
3. Why do we use trigonometric functions like cosine for problems like this?
4. How would the calculation change if the house’s base is not level (e.g., on a slope)?
5. Can the ladder length be shorter than the given base distance? Why or why not?

Tip: Always ensure your calculator is set to degrees (not radians) when solving real-world trigonometric problems involving angles measured in degrees.