To determine how much cardboard Cherese will have left over, we first need to calculate the surface area of the box she is building.
The formula for the surface area \( A \) of a rectangular box is:
\[ A = 2lw + 2lh + 2wh \]
where:
- \( l \) = length
- \( w \) = width
- \( h \) = height
Substituting the values for Cherese's box (10 inches long, 9 inches wide, and 7 inches high):
\[ A = 2(10)(9) + 2(10)(7) + 2(9)(7) \]
Calculating each term:
- \( 2(10)(9) = 180 \)
- \( 2(10)(7) = 140 \)
- \( 2(9)(7) = 126 \)
Now sum these areas:
\[ A = 180 + 140 + 126 = 446 \text{ square inches} \]
Cherese has purchased 500 square inches of cardboard. To find how much will be left over, we subtract the surface area of the box from the total amount of cardboard purchased:
\[ \text{Leftover cardboard} = 500 - 446 = 54 \text{ square inches} \]
Thus, the amount of cardboard left over is:
\[ \boxed{54 \text{ in}^2} \]