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Each 1.0 mm of a filter through which light passes reduces the intensity of the light by 12%.How thick should be to reduce the intensity of the light to


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This is a geometric series problem.

Let E be intensity of light just before entering the filter.
Each 1 mm of filter reduces intensity of light by 12% => intensity becomes 88%.

From 0mm to 1mm intensity becomes : 88% or 0.88 * E
from  1 mm to 2 mm                                        0.88 * 0.88 * E = 0.88² E
from 2 mm to 3 mm                                                = 0.88³ E

At x mm thickness intensity becomes =  0.88^x  E

finally we want  0.88^x E = 20% E = 0.20 E

 So  0.88^x = 0.20

take logarithms on both sides,

     x log 0.88 = log 0.20
     x = log 0.20 / log 0.88 = 12.59 mm

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