Question

The half-life of a certain medicine is 30 hours. The equation below gives the amount of the medicine (g, in grams) remaining in the body after t hours since taking the medicine. How many days after taking the medicine will there be 4 grams remaining in the body?

Answer 1

The number of days where 4 grams of the medicine remains in the body is

10 days.

The half-life of the medicine = 30 hours.

The equation that gives the amount of the medicine remaining in the body after t hours is:

       $g=1024~(\frac{1}{2})^{\frac{t}{30}}$

we need to calculate after how many days the medicine remaining on the body will be 4 grams.

What is half life?

Half-life is the amount of time at which the value of a quantity is reduced to half of its initial value.

The formula is given by:

       $N(t) = N_0 ~(\frac{1}{2})^{\frac{t}{t_{\frac{1}{2}}}}$

$Where,\\N(t) = amount ~after~ t~ hours\\N_0 = Initial~ amount\\t = time \\t^{\frac{1}{2}} = half~ life~ of ~the ~quantity$

We have,

g = 4 grams

Substituting in the given equation.

We get,

  $g=1024~(\frac{1}{2})^{\frac{t}{30}}\\4=1024~(\frac{1}{2})^{\frac{t}{30}}\\\frac{4}{1024}=(\frac{1}{2})^{\frac{t}{30}}\\\\\frac{1}{256}=(\frac{1}{2})^{\frac{t}{30}}$

We know that,

$2^8 = 256$

So,

$(\frac{1}{2})^8 = (\frac{1}{2})^{\frac{t}{30}}$

Since the base is the same, we can equate the exponents.

8 = t / 30.

t = 8 x 30

t = 240 hours.

24 hours = 1 day

240 hours = (240/24) days

240 hours = 10 days

Thus, we see that in 10 days the medicine will be 4 grams remaining in the body.

Learn more about half-life here:

brainly.com/question/24710827?referrer=searchResults

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