1. **State the problem:** We have a circular cake with radius $r=10$ cm. We want to cut it into identical sector pieces, each with a perimeter of approximately 23.93 cm. We need to find how many pieces the cake is cut into. 2. **Recall the formula for the perimeter of a sector:** The perimeter $P$ of a sector with radius $r$ and central angle $\theta$ (in radians) is: $$P = 2r + s$$ where $s$ is the length of the arc of the sector. 3. **Arc length formula:** The arc length $s$ is given by: $$s = r\theta$$ 4. **Express perimeter in terms of $\theta$:** $$P = 2r + r\theta = r(2 + \theta)$$ 5. **Plug in known values:** Given $P = 23.93$ cm and $r=10$ cm, $$23.93 = 10(2 + \theta)$$ 6. **Solve for $\theta$:** $$2 + \theta = \frac{23.93}{10} = 2.393$$ $$\theta = 2.393 - 2 = 0.393 \text{ radians}$$ 7. **Find the number of pieces:** The full circle has an angle of $2\pi$ radians. Number of pieces $n$ is: $$n = \frac{2\pi}{\theta} = \frac{2\pi}{0.393} \approx 16$$ 8. **Conclusion:** The cake will be cut into approximately 16 identical pieces. **Final answer:** $$\boxed{16}$$