Calculate the height of the cone.

We are given the radius ($r = 15\, \text{cm}$) and the slant height ($l = 39\, \text{cm}$). We can use the Pythagorean theorem to find the height ($h$):

$$h = \sqrt{l^2 - r^2}$$ $$h = \sqrt{(39\, \text{cm})^2 - (15\, \text{cm})^2}$$ $$h = \sqrt{1521\, \text{cm}^2 - 225\, \text{cm}^2}$$ $$h = \sqrt{1296\, \text{cm}^2}$$ $$h = 36\, \text{cm}$$

The formula for the volume of a cone is $V_{cone} = \frac{1}{3} \pi r^2 h$.
Using $\pi = 3.142$:

$$V_{cone} = \frac{1}{3} \times 3.142 \times (15\, \text{cm})^2 \times 36\, \text{cm}$$ $$V_{cone} = \frac{1}{3} \times 3.142 \times 225\, \text{cm}^2 \times 36\, \text{cm}$$ $$V_{cone} = 3.142 \times 225\, \text{cm}^2 \times 12\, \text{cm}$$ $$V_{cone} = 8483.4\, \text{cm}^3$$

The formula for the volume of a hemisphere is $V_{hemisphere} = \frac{2}{3} \pi r^3$.
Using $\pi = 3.142$ and $r = 15\, \text{cm}$:

$$V_{hemisphere} = \frac{2}{3} \times 3.142 \times (15\, \text{cm})^3$$ $$V_{hemisphere} = \frac{2}{3} \times 3.142 \times 3375\, \text{cm}^3$$ $$V_{hemisphere} = 2 \times 3.142 \times 1125\, \text{cm}^3$$ $$V_{hemisphere} = 7069.5\, \text{cm}^3$$

The total volume is the sum of the volume of the cone and the volume of the hemisphere.

$$V_{total} = V_{cone} + V_{hemisphere}$$ $$V_{total} = 8483.4\, \text{cm}^3 + 7069.5\, \text{cm}^3$$ $$V_{total} = 15552.9\, \text{cm}^3$$

The volume of the toy is $\text{15552.9 cm}^3$.