What is the greatest possible percent error in calculating the volume of a box measured to the nearest inch with sides of 5 in., 10 in., and 10 in.? round to the nearest percent

If you round to the nearest inch, the greatest possible error in every side is 0.5 in.

The greatest possible error in calculating the volume would be caused by getting the greatest possible error when measuring all the three sides (with the same sign, all positive or all negative).

Hence, If you make that greatest possible error in all three measurements, you get:

Vol. real = 5 x 10 x 10 = 500
Vol. measured = 5.5 x 10.5 x 10.5 = 606.375
Error = |Real - Measured| = 606.375 - 500 = 106.375
%Error = Error/Real = 106.375 / 500 = 0.21275 = 21.275 %

Rounded to the nearest percent, the greatest possible error is: 21%.