The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula $x = De^{-t/8}$, where $x$ is the amount of the drug in the bloodstream in milligrams and $D$ is the dose given in milligrams - Edexcel - A-Level Maths Pure - Question 2 - 2007 - Paper 6

After 5 hours, another dose of 10 mg is given, so the total dose is now $D = 20$ mg. One hour later ($t = 1$), we substitute into the formula:

$x = 20e^{-(5 + 1)/8} = 20e^{-6/8} = 20e^{-0.75}$

Calculating further:
$x = 20 \times 0.4724 \approx 9.448$

This gives the amount from the second dose alone. To find the total amount, we need to add the previous amount (after 5 h), now $5.353$ mg:

$x_{ ext{total}} = 9.448 + 5.353 = 13.801$

However, since we need the result 1 hour after the second dose, we also consider:

From the prior amount, after 5 hours: $x = 10 \cdot 10^{-1}e^{-6/8}$ which shows that the total is approximately $13.549$ mg.

We know at $t = T$, the amount of the drug in the bloodstream is $3$ mg. Using the formula:

$3 = 20e^{-T/8}$

Solving for $T$ involves first isolating $e^{-T/8}$:

$e^{-T/8} = \frac{3}{20}$

Taking the natural logarithm on both sides gives:

$-\frac{T}{8} = \ln{\left(\frac{3}{20}\right)}$

Now, multiplying by $-8$ yields:

$T = -8\ln{\left(\frac{3}{20}\right)} \approx 19.61$

Thus, the approximate value of $T$ is $19.61$ hours.