Find the volume IS A

The base is an isosceles triangle with sides $10 \text{ m}$, $10 \text{ m}$, and $12 \text{ m}$. To find its height ($h_b$), we can split it into two right-angled triangles. The base of each right triangle will be $12 \text{ m} / 2 = 6 \text{ m}$, and the hypotenuse will be $10 \text{ m}$.
Using the Pythagorean theorem:

$$h_b^2 + 6^2 = 10^2$$ $$h_b^2 + 36 = 100$$ $$h_b^2 = 64$$ $$h_b = 8 \text{ m}$$

The area of the base triangle ($A_b$) is:

$$A_b = \frac{1}{2} \times \text{base} \times h_b = \frac{1}{2} \times 12 \text{ m} \times 8 \text{ m} = 48 \text{ m}^2$$

The height of the prism ($H_{\text{prism}}$) is given as $8 \text{ m}$.

$$V_{\text{prism}} = A_b \times H_{\text{prism}} = 48 \text{ m}^2 \times 8 \text{ m} = 384 \text{ m}^3$$

The pyramid has the same triangular base. All its slant edges are given as $10 \text{ m}$. For a right pyramid with equal slant edges, the apex is directly above the circumcenter of the base.
First, find the circumradius ($R$) of the base triangle. For an isosceles triangle with equal sides $a=10 \text{ m}$ and base $b=12 \text{ m}$, and height $h_b=8 \text{ m}$:

$$R = \frac{a^2}{2h_b} = \frac{10^2}{2 \times 8} = \frac{100}{16} = \frac{25}{4} = 6.25 \text{ m}$$

Now, use the Pythagorean theorem with the slant edge ($10 \text{ m}$), the circumradius ($R$), and the pyramid height ($H_{\text{pyramid}}$):

$$H_{\text{pyramid}}^2 + R^2 = (\text{slant edge})^2$$ $$H_{\text{pyramid}}^2 + \left(\frac{25}{4}\right)^2 = 10^2$$ $$H_{\text{pyramid}}^2 + \frac{625}{16} = 100$$ $$H_{\text{pyramid}}^2 = 100 - \frac{625}{16} = \frac{1600 - 625}{16} = \frac{975}{16}$$ $$H_{\text{pyramid}} = \sqrt{\frac{975}{16}} = \frac{\sqrt{975}}{4} = \frac{\sqrt{25 \times 39}}{4} = \frac{5\sqrt{39}}{4} \text{ m}$$

$$V_{\text{pyramid}} = \frac{1}{3} \times A_b \times H_{\text{pyramid}} = \frac{1}{3} \times 48 \text{ m}^2 \times \frac{5\sqrt{39}}{4} \text{ m}$$ $$V_{\text{pyramid}} = 16 \times \frac{5\sqrt{39}}{4} \text{ m}^3 = 4 \times 5\sqrt{39} \text{ m}^3 = 20\sqrt{39} \text{ m}^3$$

The total volume is the sum of the prism's volume and the pyramid's volume.

$$V_{\text{total}} = V_{\text{prism}} + V_{\text{pyramid}}$$ $$V_{\text{total}} = 384 \text{ m}^3 + 20\sqrt{39} \text{ m}^3$$

As a numerical approximation:

$$\sqrt{39} \approx 6.245$$ $$V_{\text{total}} \approx 384 + 20 \times 6.245 = 384 + 124.9 = 508.9 \text{ m}^3$$

The volume of the solid is $384 + 20\sqrt{39} \text{ m}^3 \approx 508.9 \text{ m}^3$.