A radioactive substance has a half-life of 4 days. Suppose we have 800 mg of this substance.

a) Find the equation of the mass remaining after t days. Use the exact value for k not a decimal
approximation.
b) Rounding your answer to the nearest tenth of a mg, find the mass remaining after 30 days.
c) Rounding your answer to the nearest tenth of a day, find when the mass will be 1 mg.
d) Suppose you do not have any of this substance on hand but that you will need 1000 mg in precisely 36 hours, how many grams should you order right now to insure that you have 1000
mg in 36 hours? Round your answer up
to the “nearest” mg.

The anwser is B

Q = Qi e^-k t

when Q/Qi = 1/2
.5 = e^-k t
ln .5 = - 4 k

so k = (ln .5) / -4
Q = 800 e^ (ln .5)t / 4

k = .1733
so
Q = 800 e^-.1733 t
if t = 30
Q = 800 e^-5.2
Q = 4.4 mg

1/800 = e^-.1733 t
ln(1/800) = -.1733 t

38.6 days

enough already

a) The equation for the mass remaining after t days can be given by the formula:

M(t) = M₀ * (1/2)^(t/k),

where M(t) is the mass remaining after t days, M₀ is the initial mass, and k is the half-life of the substance.

Since the half-life is given as 4 days, we can substitute k = 4 into the equation:

M(t) = M₀ * (1/2)^(t/4).

b) To find the mass remaining after 30 days, we substitute t = 30 into the equation from part a:

M(30) = 800 * (1/2)^(30/4).

Using a calculator or computing it manually, we find M(30) to be approximately 68.8 mg (rounded to the nearest tenth of a mg).

c) To find when the mass will be 1 mg, we represent the equation from part a as follows:

1 = 800 * (1/2)^(t/4).

Solving for t using logarithms, we can take the logarithm with base 1/2 on both sides:

log₂ 1 = log₂ (800 * (1/2)^(t/4)).

Using the logarithmic property log₂ (a * b) = log₂ a + log₂ b and log₂ (a^b) = b * log₂ a, we have:

0 = log₂ 800 + (t/4) * log₂ (1/2).

Simplifying further, we get:

t/4 = -log₂ 800.

Dividing both sides by 1/4, we find:

t = -4 * log₂ 800.

Using a calculator, we find t to be approximately 35.4 days (rounded to the nearest tenth of a day).

d) To ensure having 1000 mg in 36 hours, we need to consider the decay rate of the substance to determine how much initial mass to order. We can use the equation from part a:

M(t) = M₀ * (1/2)^(t/4).

Since the desired time is 36 hours (or 1.5 days), we substitute t = 1.5 into the equation and solve for M₀:

1000 = M₀ * (1/2)^(1.5/4).

Rearranging the equation to solve for M₀, we have:

M₀ = 1000 / (1/2)^(1.5/4).

Using a calculator, we find M₀ to be approximately 3597.6 mg (rounded up to the nearest mg). Therefore, you should order 3600 mg (or 3.6 grams) in order to have 1000 mg remaining after 36 hours.