The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula $$x = De^{-0.2t}$$ where $x$ is the amount of the antibiotic in the bloodstream in milligrams, $D$ is the dose given in milligrams and $t$ is the time in hours after the antibiotic has been given - Edexcel - A-Level Maths Pure - Question 2 - 2016 - Paper 3

After the first dose, which has been given 5 hours before the second dose, the remaining amount of antibiotic is:

$x_1 = 15 e^{-0.2 \cdot 5} = 15 e^{-1} \approx 15 \cdot 0.3679 \approx 5.019$

Next, the second dose of 15 mg is administered. After 2 additional hours, the amount of the second dose in the bloodstream is:

$x_2 = 15 e^{-0.2 \cdot 2} = 15 e^{-0.4} \approx 15 \cdot 0.6703 \approx 10.055$

Thus, the total amount in the bloodstream 2 hours after the second dose is:

$x_{total} = x_1 + x_2 \approx 5.019 + 10.055 = 15.074$

However, calculating by direct substitution from the model yields:

$x_{total} = 15 e^{-0.2(5 + 2)} = 15 e^{-1.4} \approx 13.754$

Given that after 7 hours the total amount is 7.5 mg,

Assuming the equation based on previous findings:

$15 e^{-0.2(7)} + 15 e^{-0.2(2)} = 7.5$

Leading us to derive:

$T = -5 \cdot \ln \left( \frac{15}{15 + 7.5} \right) = -5 \cdot \ln \left( \frac{15}{22.5} \right)$

This expression suggests that setting $a = -5$ and determining $b$ accordingly leads us to the conclusion that $T = a \ln \left( \frac{b}{b + e} \right)$.