Question 1129782
I've been stuck trying to solve this. Could someone help explain how to correctly set up the equation. Thanks
 A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Write an exponential equation 
f(t) representing this situation. (Let f be the amount of radioactive substance in grams and t be the time in minutes.) 

The exponential equation that I thought was correct, but isn't is 250* 2( -t/h). 
<pre>Exponential GROWTH rate formula: {{{matrix(1,3, y, "=", A[o]b^t)}}}, where:
y   = REMAINING amount of substance (32, in this case)
{{{A[o]}}} = INITIAL amount of substance (250 g, in this case)
b   = GROWTH rate of substance (Unknown, in this case)
t   = time substance takes to grow (250 minutes, in this case)
{{{matrix(1,3, f(x), "=", A[o]b^t)}}}
{{{matrix(1,3, 32, "=", 250b^250)}}} ------- Substituting 32 for f(x), or y, 250 for {{{A[o]}}}, and 250 for t
{{{matrix(1,3, 32/250, "=", b^250)}}}
{{{matrix(1,3, 16/125, "=", b^250)}}} ------- Reducing fraction on left
{{{matrix(1,3, (16/125)^(1/150), "=", b^(250 * (1/250)))}}} ------- Multiplying both sides by the {{{matrix(1,2, (1/250)^th, root)}}}
{{{matrix(1,3, (16/125)^(1/250), "=", b^(cross(250) * (1/cross(250))))}}}
0.991810816, or .9918 = b
Exponential equation: {{{highlight_green(matrix(1,3, f(t), "=", 250(.9918)^t))}}}
Note: "b", being less than 1 (.9918) indicates a DECAY. If "b" were greater than 1, then that'd signify a GROWTH, as opposed to DECAY.

<b><u>OR</b></u>

As this is substance, which DECAYS/GROWS exponentially, the CONTINUOUS GROWTH/DECAY formula, or {{{matrix(1,3, A, "=", A[o]e^(kt))}}} is used.
{{{matrix(1,3, 32, "=", 250e^(250k))}}} -------- Substituting 32 for A, 250 for {{{A[o]}}}, and 250 for t
{{{matrix(1,3, 32/250, "=", e^(250k))}}}
{{{matrix(1,3, 16/125, "=", e^(250k))}}} ---------- Reducing fraction on left side
{{{matrix(1,3, 250k, "=", ln (16/125))}}} -------- Converting to NATURAL LOGARITHMIC (ln) form
k, or rate of growth/decay = {{{matrix(1,3, ln (16/125)/250, "=", - 0.0082229)}}}
As k is negative (< 0), then the substance DECAYED as opposed to it GROWING. This we already know based on the fact that initially the substance was 250 g, and reduced to 32 g in  250 hours. 
However, it's good to know that when the CONTINUOUS GROWTH/DECAY formula is used, GROWTH is indicated by a POSITIVE (> 0) value for k, and DECAY, by a NEGATIVE (< 0) value for k.

The equation for CONTINUOUS GROWTH/DECAY of a substance, or: {{{highlight(highlight_green(highlight(matrix(1,7, f(t), "=", A[o]e^(kt), "becomes:", f(t), "=", 250e^(- .008229t)))))}}}