To find the altitude $h$ of the triangle shown in the image, we will use the fact that the area of a triangle can be expressed in two different ways:

1. Using the base $a$ and the height $h$:
   $\text{Area} = \frac{1}{2} \cdot a \cdot h$

2. Using the two known sides $8$ and $15$ and the sine of the included angle $\theta$:
   $\text{Area} = \frac{1}{2} \cdot 8 \cdot 15 \cdot \sin(\theta)$

Steps to Find $h$:

1. First, calculate the hypotenuse $a$ using the Pythagorean theorem: $a = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$

2. The area of the triangle can now be expressed as: $\text{Area} = \frac{1}{2} \cdot 8 \cdot 15 = 60$

3. Use the relationship for area involving $h$: $\text{Area} = \frac{1}{2} \cdot a \cdot h$ Substituting $a = 17$ and $\text{Area} = 60$: $60 = \frac{1}{2} \cdot 17 \cdot h$

4. Solve for $h$: $h = \frac{2 \cdot 60}{17} = \frac{120}{17} \approx 7.06$

Final Answer:

The altitude $h$ is approximately $7.06$ units.

Do you want additional clarifications or have any other questions?

Here are 5 related questions to expand on this topic:

1. How can the sine function help determine other elements in this triangle?
2. What is the value of $\theta$ in the triangle using trigonometry?
3. How can we verify the calculation of $h$ using another method?
4. What are the other two altitudes of this triangle?
5. How can you calculate the area using other trigonometric ratios?

Tip: In right triangles, always use the Pythagorean theorem to find missing sides before exploring other properties!