A cone has height 24 cm and base radius 10 cm…

Step 1. The small cone. Its height is $24\ \text{cm} - 9\ \text{cm} = 15\ \text{cm}$.

Step 2. The radius of the cut. $\dfrac{r}{10\ \text{cm}} = \dfrac{15\ \text{cm}}{24\ \text{cm}}$, so $r = 6.25\ \text{cm}$.

Step 3. The slant height. $l = \sqrt{9^2 + (10 - 6.25)^2}\ \text{cm} = \sqrt{81 + 14.0625}\ \text{cm} = \sqrt{95.0625}\ \text{cm} = 9.75\ \text{cm}$.