Initially 100 milligrams of a radioactive substance was present. after 8 hours the mass had decreased by 6%. if the rate of decay is proportional to the amount of the substance present at time t, determine the half-life of the radioactive substance.

  89.6 hours

Step-by-step explanation:

Each hour, the initial quantity is multiplied by (1 -0.06)^(1/8), so after t hours, the multiplier is 0.94^(t/8). We want to find the value of t that makes this multiplier equal to 1/2.

  0.5 = 0.94^(t/8)

  log(0.5) = (t/8)log(0.94) . . . . . . . . take logs

  t = 8·log(0.5)/log(0.94) ≈ 89.6 . . . divide by the coefficient of t

The half-life of the substance is 89.6 hours.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4