The half-life of a certain radioactive material is 38 days. an initial amount of the material has a mass of 497 kg. write an exponential function that models the decay of this material. find how much radioactive material remains after 4 days. round your answer to the nearest thousandth. (1 point) a ; 462.029 kg b ; 0.087 kg c ; 1.040 kg d ; 0 kg

It is the 1st equation at the top. 

Reason: First check the equations to check that the initial amount is 497 kg. You can do this by setting x = 0 into all of the equations. The 3rd and 4th equations evaluate to 2 when x = 0 and so you can eliminate the bottom 2 equations immediately. Equation # 2 does not work since the half-life value of 1.040 kg is way to small (significantly smaller than half of 497 kg). 

You can check that equation # 1 is the right one, by setting x = 0 and getting 

y = 497*(1/2)^[(1/38) * 0] = 497 

so the initial amount is 497 

Also check that there is 1/2 the amount at time 38 (since the half-life is 38 days) 

y = 497 * (1/2)^[(1/38) * 38] = 248.5 

248.5 kg is half of 497 and so this checks out for equation # 1. 

Since we know equation # 1 is good, now we evaluate at x = 4 to get 

y = 497 * (1/2)^[(1/38) * 4] = 462.029 

so our answer to the thousandth places is 462.029 kg.