Alex is given a 15 mg injection of the drug at the same time every day for a long period of time - Leaving Cert Mathematics - Question 9 - 2022

To find the total amount after the 10th injection, we use the formula for the sum of a geometric series. The amount in Alex’s body after n injections is given by:

S_n = rac{a(1 - r^n)}{1 - r}

where:

• $a = 15$ (the initial amount),
• $r = 0.6$ (the ratio of the remaining drug) and,
• $n = 10$.

Thus:

S_{10} = rac{15(1 - (0.6)^{10})}{1 - 0.6} = rac{15(1 - 0.006046619)}{0.4}

Now, calculating:

S_{10} = rac{15(0.993953381)}{0.4} ext{ which results in approximately } 37.27 ext{ mg.}

This gives the total amount of the drug in Alex's body immediately after the 10th injection as 37.27 mg.

To find the amount of the drug in Alex's body after a long period of daily injections, we apply the sum to infinity of a geometric series, which is given by:

S_{ ext{infinity}} = rac{a}{1 - r}

Given:

• $a = 15$ mg (amount from each injection),
• $r = 0.6$ (because 40% is eliminated each day).

Thus:

S_{ ext{infinity}} = rac{15}{1 - 0.6} = rac{15}{0.4} = 37.5 ext{ mg}

Therefore, the estimated total amount of the drug in Alex’s body after a long period is approximately 37.5 mg.

The expression for the total amount immediately after the i-th injection, given that each day Jessica receives an amount that decreases by 15%, can be formulated as:

$d + d(1 - 0.85) + ... + d(1 - 0.85)^{(i-1)}$

This series represents every injection Jessica has received, compounded by the reduction of drug left in her body from prior injections. Here:

• $d$ is the initial amount,
• $1 - 0.85$ is used to indicate the amount that remains after each injection.

It represents a finite geometric series.

Given that after the 7th injection there are 50 mg in Jessica’s body, we can use the previous geometric series form to solve for $d$:

$d(1 - 0.85^7)/(1 - 0.85) = 50$

Now substituting the values, we get:

$d(1 - 0.85^7) = 50(0.15)$

This simplifies to:

$d(1 - 0.19857899) = 50(0.15)$

Calculating gives:

ightarrow d = rac{7.5}{0.80142101}$$ Thus: $$d ≈ 9.34 ext{ mg (to nearest mg: } 9 ext{ mg)}$$ Therefore, the amount of drug in one of Jessica’s daily injections is approximately 9 mg.