Question

The table below illustrates the decay of a sample of radioactive uranium. Time in Days, x 0 1 2 3 4 5 Sample Remaining (grams), U 500 255 130 66 34 17 Which equation best models this set of data where U represents the amount of sample remaining, in grams, at time x?

Answer 1

`\underline\ \ 5\ \ \\U|500\ |\ 255|\ 130|\ 66\ \ |\ 34\ |\ \ 17\ |\\\\U=a(b)^x\\\\for\ x=0,\ U=500\\\\500=a(b)^0\\\\500=a(1)\to \boxed{a=500}\\\\for\ x=1,\ U=255\\\\255=500(b)^1\\\\255=500b\qquad\text{divide both sides by 500}\\\\b=(255)/(500)\\\\b=(255:5)/(500:5)\\\\b=(51)/(100)\to \boxed{b=0.51}\\\\\text{Therefore we have the equation of the function:}\\\\U=500(0.51)^x`

`\text{Check for other values of x:}\\\\for\ x=2\\\\U=500(0.51)^2=130.05\approx130\qquad CORRECT\\\\for\ x=3\\\\U=500(0.51)^3=66.3255\approx66\qquad CORRECT\\\\for\ x=4\\\\U=500(0.51)^4=33.826\approx34\qquad CORRECT\\\\for\ x=5\\\\U=500(0.51)^5=17.25125\approx17\qquad CORRECT`

`Answer:\ \boxed{U=500(0.51)^x}`