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 equation you want is (1/2^t/4)= x amount remaining which in this case is (1/2^30/4) = (1/2^7.5)= x

Solve that equation for the amount remaining after 30 days.
Using that equation you should be able to do the others. For example c is
[(1/2^(t/4)] = 1 and solve for t.

To solve this problem, we can use the formula for the decay of a radioactive substance, which is given by:

M(t) = M₀ * (1/2)^(t/T)

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

a) In this case, the initial mass is 800 mg and the half-life is 4 days. Therefore, the equation for the mass remaining after t days is:

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

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

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

To round the answer to the nearest tenth of a mg, we can evaluate this expression on a calculator or using a suitable software. The result will be the mass remaining after 30 days.

c) To find when the mass will be 1 mg, we set M(t) = 1 and solve for t:

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

To round the answer to the nearest tenth of a day, we can use logarithms to isolate t. Taking the logarithm to the base 2 of both sides of the equation, we get:

log₂(1) = log₂(800 * (1/2)^(t/4))

0 = log₂(800) + log₂((1/2)^(t/4))

0 = log₂(800) + (t/4) * log₂(1/2)

0 = log₂(800) - (t/4)

t/4 = log₂(800)

t = 4 * log₂(800)

Again, we can use a calculator or suitable software to evaluate this expression and round the answer to the nearest tenth of a day.

d) To calculate how many grams should be ordered, we need to consider the decay rate. The decay rate can be calculated using the half-life. In this case, the half-life is 4 days.

First, we need to convert the time frame from hours to days. Since there are 24 hours in a day, 36 hours is equal to 36/24 = 1.5 days.

Next, we can use the decay rate to determine the remaining mass after 1.5 days:

M(1.5) = 800 * (1/2)^(1.5/4)

This will give us the mass remaining after 1.5 days.

Finally, we subtract the remaining mass from 1000 mg to find how much more is needed:

1000 - M(1.5) = additional mass needed

To ensure that we have at least 1000 mg in 36 hours, we round up the additional mass needed to the nearest mg.