To find out how much cardboard Cherese will have left over, we need to calculate the surface area of the box she is building and then subtract that from the amount of cardboard she purchases.
The formula for the surface area \(A\) of a rectangular box is given by:
\[ A = 2(lw + lh + wh) \]
where:
- \(l\) is the length,
- \(w\) is the width,
- \(h\) is the height.
Given dimensions are:
- \(l = 10\) inches,
- \(w = 9\) inches,
- \(h = 7\) inches.
Now plugging in the values:
\[ A = 2(10 \cdot 9 + 10 \cdot 7 + 9 \cdot 7) \]
Calculating each term inside the parentheses:
- \(10 \cdot 9 = 90\)
- \(10 \cdot 7 = 70\)
- \(9 \cdot 7 = 63\)
Now, sum these up:
\[ 90 + 70 + 63 = 223 \]
Now, multiply by 2 for the surface area:
\[ A = 2 \cdot 223 = 446 \text{ square inches} \]
Now, Cherese purchased 500 square inches of cardboard. To find out how much will be left over, we subtract the surface area of the box from the total amount of cardboard:
\[ \text{Left over} = 500 - 446 = 54 \text{ square inches} \]
Therefore, the amount of cardboard left over is:
\[ \boxed{54 \text{ in. squared}} \]