We are given the following problem about a solid with triangular cross-sections:

1. Base: The base is the region bounded by the curves $y = 5 - \frac{5}{9}x^2$ and $y = 0$ (the x-axis).
2. Cross-sections: Every cross-section parallel to the y-axis is an isosceles triangle where the height and base of each triangle are equal.

Steps to solve:

To find the volume, we will:

1. Identify the bounds of integration by finding where the curves intersect the x-axis (i.e., $y = 0$).
2. Use the formula for the area of a cross-sectional triangle, which depends on the function $y = 5 - \frac{5}{9}x^2$.
3. Integrate this area over the appropriate limits to find the volume.

Step 1: Find the limits of integration

The base is bounded by $y = 5 - \frac{5}{9}x^2$ and $y = 0$. Set $y = 0$ to find the x-values where the parabola intersects the x-axis:

$0 = 5 - \frac{5}{9}x^2$

Solve for $x$:

$\frac{5}{9}x^2 = 5 \quad \Rightarrow \quad x^2 = 9 \quad \Rightarrow \quad x = \pm 3$

So, the limits of integration are from $x = -3$ to $x = 3$.

Step 2: Area of cross-sections

The height and base of each triangle are given to be equal, so if the base length is $y$, then the area of the triangle is:

$A(x) = \frac{1}{2} \cdot y \cdot y = \frac{1}{2} y^2$

Substitute $y = 5 - \frac{5}{9}x^2$:

$A(x) = \frac{1}{2} \left(5 - \frac{5}{9}x^2\right)^2$

Step 3: Integrate to find the volume

We now integrate the area function over the limits $x = -3$ to $x = 3$:

$V = \int_{-3}^{3} \frac{1}{2} \left(5 - \frac{5}{9}x^2\right)^2 dx$

Let's calculate this integral. I will proceed with the integration now.The volume of the solid is $40$ cubic units.

Would you like further details on any part of the solution? Here are some related questions to explore:

1. How do we compute the area of different cross-sections for other shapes?
2. What would happen if the height and base of the triangles were not equal?
3. How does the method change if the base is bounded by different curves?
4. What other cross-sectional shapes could be used, and how would that affect the integration process?
5. What is the geometric significance of the function $y = 5 - \frac{5}{9}x^2$?

Tip: For solids with known cross-sectional shapes, the area formula plays a crucial role in finding the volume.