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)=(.98)^t

(b) Estimate the half-life of strontium.

(c) If presently there is 6 grams of strontium, estimate how many grams of the substance will remain after 38 years.

i need help with b and c

a)

P = (.98)^t , you had that

b)
1/2 = .98^t
.5 = .98^t
take logs of both sides and use log rules ...
log .5 = t(log .98)
t = log .5/log .98 = appr 34.4 years

c) P = 6(.98)^38
= aprr 2.78 g

makes sense, since the half-life of 34.4 years would have reduced it to 3 g , and we went just a few more years.

write a formula,p(t), that models the percent of strontium-90 after t years.

:
The radioactive decay formula
A = Ao*2^(-t/h)
Where
A = resulting amt after t time
Ao - initial amt
h = half-life of substance
t = time of decay
:
To find percent here, you could write it
p(t) = 100*2^(-t/28)

Sure, I'd be happy to help with parts (b) and (c)!

(b) To estimate the half-life of strontium, we need to find the time it takes for half of the substance to decay. In this case, we know that the decay rate is 2 percent per year. So let's set up an equation to find the half-life:

(0.98)^t = 0.5

To solve for t, we can take the logarithm of both sides of the equation:

t * log(0.98) = log(0.5)

Now, we can solve for t:

t = log(0.5) / log(0.98)

Using a calculator, we find that t ≈ 35.16 years. So, the estimated half-life of strontium is approximately 35.16 years.

(c) To estimate how many grams of strontium will remain after 38 years, we can use the fraction P(t) = (.98)^t that we derived in part (a). We know that initially there were 6 grams of strontium, so we can plug in t = 38 into the equation to find the remaining amount:

P(38) = (.98)^38

Using a calculator, we find that P(38) ≈ 3.1115 grams. Therefore, it is estimated that approximately 3.1115 grams of strontium will remain after 38 years.

Remember, these estimates assume the decay rate remains constant at 2 percent per year.

(b) To estimate the half-life of strontium, we can solve the equation P(t) = 0.5 for t, where P(t) is the fraction remaining at time t.

0.5 = (.98)^t

To solve this equation, we can take the logarithm of both sides:

log(0.5) = log((.98)^t)

Since log((.98)^t) can be rewritten as t * log(.98), we have:

log(0.5) = t * log(.98)

Now we can solve for t using basic algebraic steps:

t = log(0.5) / log(.98)

Using a scientific calculator, we find:

t ≈ 34.77 years

Therefore, the estimated half-life of strontium is approximately 34.77 years.

(c) To estimate how many grams of strontium will remain after 38 years, we can use the fraction P(t) = (.98)^t and substitute t = 38 into the equation:

P(38) = (.98)^38

Using a calculator, we find:

P(38) ≈ 0.472

To find the amount of strontium remaining, we multiply the fraction by the initial amount (6 grams):

Amount remaining = 0.472 * 6

Therefore, the estimate for the amount of strontium remaining after 38 years is approximately 2.832 grams.

To estimate the half-life of strontium, we can use the formula:

P(t) = (0.5)^n

Where n is the number of half-lives. Since the decay rate of strontium-90 is 2 percent per year, we can calculate the half-life by finding the number of years it takes for the remaining fraction to reach 0.5.

We'll solve for n in the equation:

(0.98)^t = 0.5

Taking the natural logarithm of both sides:

ln((0.98)^t) = ln(0.5)

Using the logarithmic property:

t * ln(0.98) = ln(0.5)

Now divide both sides by ln(0.98) to solve for t:

t = ln(0.5) / ln(0.98)

Using a calculator, we can find:

t ≈ -34.985 years

Since time cannot be negative in this context, we take the absolute value:

t ≈ 34.985 years

This value represents the approximate half-life of strontium-90.

Moving on to part (c), we can use the formula P(t) = (0.98)^t to estimate how many grams of strontium will remain after 38 years.

P(38) = (0.98)^38

Using a calculator, we can determine:

P(38) ≈ 0.47

This means that approximately 47% of the strontium will remain after 38 years.

To estimate the number of grams remaining, we multiply the percentage by the initial amount of strontium (6 grams):

0.47 * 6 ≈ 2.82 grams

Therefore, it is estimated that around 2.82 grams of strontium will remain after 38 years.