a radioactive substance decays at an annual rate of 24 percent. If the initial amount of the substance is 640 grams, which of the following functions (f) models the remaining amount of the substance, in grams, t years later?
a) f(t) = 640(0.76)^t
b) f(t) = 0.76(640)^t
c) f(t)= 0.24(640)^t
each year, the amount remaining is 76% of the previous year ... 100% - 24%
To find the correct function that models the remaining amount of the substance, we need to understand the concept of exponential decay and how to solve this problem step by step.
First, let's break down the problem. We know that the radioactive substance decays at an annual rate of 24 percent. This means that each year, 24 percent of the remaining substance will decay.
To find the remaining amount of the substance after a certain number of years, we need to multiply the initial amount by the rate of decay each year.
Here's how to find the remaining amount of the substance using the given information:
1. Start with the initial amount: 640 grams.
2. Calculate the decay rate: 24 percent = 0.24.
3. Multiply the initial amount by the decay rate: 640 * 0.24 = 153.6 grams.
4. Subtract the decayed amount from the initial amount: 640 - 153.6 = 486.4 grams.
5. Repeat steps 3 and 4 for each additional year.
Now, let's look at the functions provided and identify the correct one:
a) f(t) = 640(0.76)^t
This function reflects a decrease of 76%, not the correct 24%.
b) f(t) = 0.76(640)^t
Similarly, this function also reflects a decrease of 76%, not the correct 24%.
c) f(t)= 0.24(640)^t
This function correctly represents the decay rate of 24%. It multiplies the initial amount (640 grams) by the decay rate (0.24) raised to the power of time (t).
Therefore, the correct function that models the remaining amount of the substance t years later is option c:
f(t)= 0.24(640)^t