Suppose strontium-90 decays at a rate of 2 percent per year. (a) Write the fraction P of strontium remaining, as function of t, measured in years. (Assume that at time t=0 there is 100 % remaining.)

Answer: P(t) = (b) Estimate the half-life of strontium.
Answer:
Hint: Use your graphing calculator and the trace function. (Or use natural logarithms as in 1.9).(c) If presently there is 5.5 grams of strontium, estimate how many grams of the substance will remain after 11 years.
Answer:

(a) To write the fraction P of strontium remaining as a function of time t, we can start by considering that each year, the strontium decays at a rate of 2%. This means that the remaining amount is equal to 100% (or 1 as a fraction) minus 2% (or 0.02 as a decimal) for each year that passes.

So, for each year, the remaining fraction P(t) can be written as:
P(t) = 1 - 0.02t

(b) To estimate the half-life of strontium, we need to find the time it takes for the remaining fraction to decrease to half of its original value, which is 0.5.

Using the equation from part (a), we can set P(t) equal to 0.5 and solve for t:
0.5 = 1 - 0.02t

Rearranging the equation, we get:
0.02t = 1 - 0.5
0.02t = 0.5

Dividing both sides by 0.02, we get:
t = 0.5 / 0.02
t ≈ 25

So, the estimated half-life of strontium is 25 years.

(c) To estimate how many grams of strontium will remain after 11 years, we can use the equation from part (a) and substitute t = 11:
P(11) = 1 - 0.02 * 11

Simplifying the equation, we get:
P(11) = 1 - 0.22
P(11) ≈ 0.78

To find the remaining amount in grams, we can multiply the fraction P(11) by the initial amount of strontium (which is 5.5 grams):
Remainder = P(11) * 5.5
Remainder ≈ 0.78 * 5.5
Remainder ≈ 4.29

So, after 11 years, it is estimated that approximately 4.29 grams of strontium will remain.