The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula $$x = De^{-\frac{1}{8} t}$$, 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 3 - 2007 - Paper 5

After the second dose of 10 mg is given 5 hours later, the total dose in the bloodstream can be calculated using:

$D = 10 + 10 e^{-\frac{1}{8} \cdot 5}$

Now, we need to find the amount of the drug in the bloodstream 1 hour after the second dose (i.e., at $t = 6$):

$x = D e^{-\frac{1}{8} \cdot 1}$ $x = (10 + 5.353)e^{-\frac{1}{8}}$

Calculating this gives: $x = 15.353 e^{-0.125}$

Using a calculator to find $e^{-0.125}$, we have: $x \approx 15.353 \cdot 0.882 = 13.549$

Thus, the amount of the drug in the bloodstream 1 hour after the second dose is 13.549 mg.

To find the value of T when the amount in the bloodstream is 3 mg, we set up the equation:

$3 = 10 e^{-\frac{1}{8} (5 + T)}$

Rearranging gives: $e^{-\frac{1}{8} (5 + T)} = \frac{3}{10}$

Taking the natural logarithm on both sides gives: $-\frac{1}{8}(5 + T) = \ln(\frac{3}{10})$

Thus, $5 + T = -8 \ln(\frac{3}{10})$

Calculating the right-hand side: $T = -8 \ln(\frac{3}{10}) - 5$

This solution represents the time T when the drug amount is 3 mg.