Calculate the height of the trapezoidal base.

From the diagram, we can form a right-angled triangle within the trapezoid. The hypotenuse is the slanted side (5 m), and one leg is the horizontal difference between the parallel sides (implied as 4 m from the student's calculation $5^2 - 4^2$). Let $h_{trap}$ be the height of the trapezoid.
Using the Pythagorean theorem:

$$h_{trap}^2 + 4^2 = 5^2$$ $$h_{trap}^2 + 16 = 25$$ $$h_{trap}^2 = 25 - 16$$ $$h_{trap}^2 = 9$$ $$h_{trap} = \sqrt{9}$$ $$h_{trap} = 3 \text{ m}$$

One parallel side ($b_1$) is given as 3 m.
The other parallel side ($b_2$) is the sum of $b_1$ and the horizontal difference:

$$b_2 = 3 \text{ m} + 4 \text{ m} = 7 \text{ m}$$

So, the parallel sides are 3 m and 7 m.

The formula for the area of a trapezoid is $A = \frac{1}{2} (b_1 + b_2) h_{trap}$.

$$A_{base} = \frac{1}{2} (3 \text{ m} + 7 \text{ m}) \times 3 \text{ m}$$ $$A_{base} = \frac{1}{2} (10 \text{ m}) \times 3 \text{ m}$$ $$A_{base} = 5 \text{ m} \times 3 \text{ m}$$ $$A_{base} = 15 \text{ m}^2$$

The formula for the volume of a prism is $V = A_{base} \times L$, where $L$ is the length of the prism.
From the diagram, the length of the prism is 15 m.

$$V = 15 \text{ m}^2 \times 15 \text{ m}$$ $$V = 225 \text{ m}^3$$

The volume of the prism is $\text{225 m}^3$.