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).
Exponential GROWTH rate formula: , where:
y = REMAINING amount of substance (32, in this case)
= 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)
------- Substituting 32 for f(x), or y, 250 for , and 250 for t
------- Reducing fraction on left
------- Multiplying both sides by the
0.991810816, or .9918 = b
Exponential equation:
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.
OR
As this is substance, which DECAYS/GROWS exponentially, the CONTINUOUS GROWTH/DECAY formula, or is used.
-------- Substituting 32 for A, 250 for , and 250 for t
---------- Reducing fraction on left side
-------- Converting to NATURAL LOGARITHMIC (ln) form
k, or rate of growth/decay =
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: