The manufacturer of a laser printer reports the mean number

The manufacturer of a laser printer reports the mean number of pages a cartridge will print before it needs replacing is 12,300. The distribution of pages printed per cartridge closely follows the normal probability distribution and the standard deviation is 720 pages. The manufacturer wants to provide guidelines to potential customers as to how long they can expect a cartridge to last. How many pages should the manufacturer advertise for each cartridge if it wants to be correct 90 percent of the time? (Round z value to 2 decimal places. Round your answer to the nearest whole number.)

i understand that z = x - 12300 / 720. and ive found answers saying If 90 percent of the values are above x, then the area between x and the mean is 0.4900. Search the z table for the area closest to 0.3997. It is 0.3997. Move to the margins from this value to find z is –1.28. But how are computing .3997? I know how to solve the problem if i have the z value but i dont know how to derive the z value from the equation z = x - 12300 / 720 or 720(z) = x-12300

Solution

First, we get the z score from the given left tailed area. As          
          
Left tailed area = 1 - 0.9 = 0.1      
          
Then, using table or technology,          
          
z =    -1.281551566      
          
As x = u + z * s,          
          
where          
          
u = mean =    12300      
z = the critical z score =    -1.281551566      
s = standard deviation =    720      
          
Then          
          
x = critical value =    11377.28287 = 11377 [ANSWER]