To solve the problem of determining the length of the ladder, we use trigonometry. The setup of the problem involves a right triangle where:

• The ladder is the hypotenuse.
• The vertical height the ladder reaches up the wall is the opposite side (8.5 feet).
• The angle of elevation is $60^\circ$.

The trigonometric function that relates the opposite side and the hypotenuse is sine:

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Step 1: Substitute the given values

$\sin(60^\circ) = \frac{8.5}{\text{length of the ladder}}$

Step 2: Solve for the length of the ladder

Rearranging the equation: $\text{length of the ladder} = \frac{8.5}{\sin(60^\circ)}$

The value of $\sin(60^\circ)$ is a known trigonometric constant: $\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$

Substitute: $\text{length of the ladder} = \frac{8.5}{0.866} \approx 9.82 \, \text{feet}$

Final Answer:

The length of the ladder is approximately 9.82 feet.

Would you like further clarification or details about any part of this calculation? Here are related questions to expand your understanding:

1. How is the sine function derived in a right triangle?
2. What are some other trigonometric functions and when are they used?
3. How do we calculate trigonometric values without a calculator?
4. What would change if the angle of elevation were smaller?
5. How can this calculation be visualized as part of a real-world application?

Tip: Always verify that your triangle is a right triangle before applying basic trigonometric functions!