To find the height of the box of cereal, we need to use the formula for the volume of a rectangular box:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
We are given the volume of the box as \( 10,000 , \text{cm}^3 \) and the dimensions of the base as \( 25 , \text{cm} \) (length) and \( 10 , \text{cm} \) (width).
First, we calculate the area of the base:
\[ \text{Base Area} = \text{Length} \times \text{Width} = 25 , \text{cm} \times 10 , \text{cm} = 250 , \text{cm}^2 \]
Now, we can plug in the volume and the base area into the volume formula to find the height:
\[ 10,000 , \text{cm}^3 = 250 , \text{cm}^2 \times \text{Height} \]
To find the height, we can rearrange the formula:
\[ \text{Height} = \frac{\text{Volume}}{\text{Base Area}} = \frac{10,000 , \text{cm}^3}{250 , \text{cm}^2} \]
Now, we perform the division:
\[ \text{Height} = \frac{10,000}{250} = 40 , \text{cm} \]
Thus, the height of the box of cereal is
\[ \boxed{40 , \text{cm}} \]