Start from $40$ grams and figure out how much remains after 4 days, then after 8 days, then after 12 days, and so on.

Keep halving the amount every 4 days until you reach exactly $5$ grams. How many 4-day periods did that take, and what is the total time?

In Desmos, type the function for the amount after $t$ days: y = 40*(1/2)^(x/4) (use x for time in days). This represents repeated halving every 4 days.

On a new line, type y = 5 to draw a horizontal line showing where the amount is $5$ grams.

Use the intersection tool (or tap where the curves cross) to find the point where y = 40*(1/2)^(x/4) meets y = 5. The x-coordinate of this intersection is the number of days it takes for the sample to decay to 5 grams.

A half-life of $4$ days means that every 4 days, the amount of the substance is cut in half. So after each 4-day interval, you multiply the current amount by $1/2$ (or divide by $2$).

Start with the initial amount of $40$ grams and apply the half-life repeatedly, keeping track of time:

• At the start (day $0$): $40$ grams
• After $4$ days (one half-life): $40 \div 2 = 20$ grams
• After $8$ days (two half-lives): $20 \div 2 = 10$ grams
• After the next half-life: $10 \div 2 = 5$ grams

We now see the next halving brings the amount to $5$ grams.

From the list:

• It takes three half-lives to go from $40$ grams to $5$ grams.
• Each half-life is $4$ days, so the total time is

So the correct answer is 12 days.