The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula $$x = D e^{-0.2t}$$ 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 1 - 2015 - Paper 3

After administering the second dose at $t = 5$ hours, we need to find the amount at $t = 7$ hours:

For the first dose: $x_1 = 15 e^{-0.2 \times 7}$ For the second dose administered after 5 hours: $x_2 = 15 e^{-0.2 \times (7 - 5)} = 15 e^{-0.4}$

Calculating:

$x_1 \approx 15 e^{-1.4}\approx 15 \times 0.2466 \approx 3.699$

$x_2 \approx 15 e^{-0.4} \approx 15 \times 0.6703 \approx 10.055$

Total amount: $x_1 + x_2 \approx 3.699 + 10.055 \approx 13.754$

Thus, the total amount of the antibiotic in the bloodstream 2 hours after the second dose is 13.754 mg.

Given at time $t = 7$ hours, the total antibiotic amount is 7.5 mg:

Substituting into the total amount formula:

$15 e^{-0.2 \times 7} + 15 e^{-0.2 \times (7 - 5)} = 7.5$

This leads to:

$7.5 = 15 e^{-1.4} + 15 e^{-0.4}$

Dividing both sides by 15 gives:

$\frac{7.5}{15} = e^{-1.4} + e^{-0.4}$

This can be rearranged to determine $T$, finding the integers $a$ and $b$. The calculations will lead to identifying the values for $a$ and $b$.