放射性物质 $Q$ 每分钟以其质量的 6% 的恒定速率衰变。如果 $M$ 是 $Q$ 的初始质量，解释为什么 $t$ 分钟后 $Q$ 的质量为 $(0.94)^t M$。 10 分钟后，$Q$ 的 $x$% 发生了衰变。求 $x$ 的值（保留 2 位有效数字）。 已知 $t$ 分钟后剩余的 $Q$ 的质量可以表示为 $M e^{kt}$，求常数 $k$ 的值。

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.

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