A radioactive substance $Q$ decays at a constant rate of 6% of its mass every minute. If $M$ is the initial mass of $Q$, explain why the mass of $Q$ after $t$ minute is $(0.94)^t M$. $x$% of $Q$ decayed after 10 minutes. Calculate, to 2 significant figures, the value of $x$. Given that the mass of $Q$ left after $t$ minutes can be expressed as $M e^{kt}$, find the value of the constant $k$.

1. Find the mass of $Q$ after 1 minute

Initial mass = $M$
Mass after 1 minute = $M - 0.06M = 0.94M$

2. Generalise the mass of $Q$ after $t$ minutes

Mass after 2 minutes = $0.94M - 0.06(0.94M) = 0.94(0.94M) = (0.94)^2 M$
Since the mass is multiplied by $0.94$ every minute, the mass after $t$ minutes is $(0.94)^t M$.

1. Find the proportion of the mass remaining after 10 minutes

Mass remaining after 10 minutes $= (0.94)^{10}M \approx 0.538615M$

2. Calculate the percentage decayed and round to 2 significant figures

Mass decayed $= M - 0.538615M = 0.461385M$
Percentage decayed $= 0.461385 \times 100\% = 46.1385\%$
$x = 46$ (to 2 s.f.)

1. Equate the two expressions for the mass of $Q$ after $t$ minutes.

$M e^{kt} = M(0.94)^t$

2. Solve for $k$ by taking the natural logarithm.

$e^{kt} = (0.94)^t$
$e^k = 0.94$
$k = \ln(0.94)$
$k \approx -0.061875 \approx -0.0619 \text{ (to 3 s.f.)}$