To find out how much cardboard Cherese will have left over, we first need to calculate the surface area of the box she wants to build.
The dimensions of the box are:
- Length (l) = 10 inches
- Width (w) = 9 inches
- Height (h) = 7 inches
The formula to calculate the surface area (SA) of a rectangular box is:
\[ SA = 2(lw + lh + wh) \]
Now we can plug in the values:
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Calculate \(lw\): \[ lw = 10 \times 9 = 90 , \text{in}^2 \]
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Calculate \(lh\): \[ lh = 10 \times 7 = 70 , \text{in}^2 \]
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Calculate \(wh\): \[ wh = 9 \times 7 = 63 , \text{in}^2 \]
Now, adding these areas together: \[ lw + lh + wh = 90 + 70 + 63 = 223 , \text{in}^2 \]
Now, multiply by 2 to get the total surface area: \[ SA = 2 \times 223 = 446 , \text{in}^2 \]
Cherese purchases 500 square inches of cardboard. To find out how much cardboard she will have left over, we subtract the surface area of the box from the total amount of cardboard she purchased:
\[ \text{Leftover} = 500 - 446 = 54 , \text{in}^2 \]
Thus, the amount of cardboard left over is \(\boxed{54 , \text{in}^2}\).