math word problems

Question

(1) A radioactive substance decays according to the following function, where y0 is the initial amount present, and y is the amount present at time t (in days).

y=y₀e^−0.092t

Find the half-life of this substance. Do not round any intermediate computations, and round your answer to the nearest tenth.

(2)A city has a population of 380,000 people. Suppose that each year the population grows by 8.5%. What will the population be after 13 years?

Answer 1

Hello Lily!

a)

The half-life is when the substance reach half of the initial value, in other words:

y = y₀/2

So we have:

y/y₀ = 2^-1 = (e^ln2)^-1

(e^ln2)^-1 = e^-0.092t

0.092t = ln2

t = 7.5

b)

Each year the population grows 8.5% of the previous year. So in the first year we'll have:

380,000 + 380,000*8,5% = 380,000*(1+8,5%)

In the second year, we'll have:

380,000*(1+8,5%) +380,000*(1+8,5%)*8,5% = 380,000*(1+8,5%)²

The formula for the population is given as:

380,000*(1+8,5%)ⁿ , where n is the number of years.

So in 13 years, the population will be:

380,000*(1+8,5%)¹³ = 1,097,413

Feel free to ask any question! Best wishes!