Question

The half life of a radioactive substance is the time it takes for a quantity of the substance to decay to half of the initial amount . The half-life of the radioactive gas radon is approximately 3.8 days. The initial amount of radon used in an experiment is 75 grams. if N represents the number of grams of radon remaining t days after the start of the experiment,
a. Write an equation that gives N in terms of t.
b. How much gas radon approximately remains after 3.8 days?
c. approximately when will the amount of radon remaining be 10 grams?

Answer 1

Using an exponential function, it is found that:

a) $N(t) = 75(0.5)^{\frac{t}{3.8}}$

b) 37.5 grams of the gas remains after 3.8 days.

c) The amount remaining will be of 10 grams after approximately 11 days.

What is an exponential function?

A decaying exponential function is modeled by:

$A(t) = A(0)(1 - r)^t$

In which:

• A(0) is the initial value.

• r is the decay rate, as a decimal.

Item a:

We start with 75 grams, and then work with a half-life of 3.8 days, hence the amount after t daus is given by:

$N(t) = 75(0.5)^{\frac{t}{3.8}}$

Item b:

This is N when t = 3.8, hence:

$N(t) = 75(0.5)^{\frac{3.8}{3.8}} = 37.5$

37.5 grams of the gas remains after 3.8 days.

Item c:

This is t for which N(t) = 10, hence:

$N(t) = 75(0.5)^{\frac{t}{3.8}}$

$10 = 75(0.5)^{\frac{t}{3.8}}$

$(0.5)^{\frac{t}{3.8}} = \frac{10}{75}$

$\log{(0.5)^{\frac{t}{3.8}}} = \log{\frac{10}{75}}$

$\frac{t}{3.8}\log{0.5} = \log{\frac{10}{75}}$

$t = 3.8\frac{\log{\frac{10}{75}}}{\log{0.5}}$

$t \approx 11$

The amount remaining will be of 10 grams after approximately 11 days.

More can be learned about exponential functions at brainly.com/question/25537936