## Question 9:

### Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Subs $D=15$ and $t=4$: $x = 15e^{-0.2\times4} = 6.740\ (mg)$ | M1 | Attempts to substitute both $D=15$ and $t=4$ in $x = De^{-0.2t}$. Can be implied by sight of $15e^{-0.8}$, $15e^{-0.2\times4}$ or awrt 6.7. Condone slips on the power. |
| | A1 | CAO $6.740\ (mg)$. Note $6.74\ (mg)$ is A0 |

### Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $15e^{-0.2\times7} + 15e^{-0.2\times2} = 13.754\ (mg)$ | M1 | Attempt to find sum of two expressions with $D=15$ in both terms with $t$ values of 2 and 7. Evidence: $15e^{-0.2\times7}+15e^{-0.2\times2}$ or $(15e^{-1}+15)e^{-0.2\times2}$. Award for sight of the two numbers awrt **3.70** and awrt **10.05** followed by total awrt **13.75** |
| | A1* | cso — both the expression $15e^{-0.2\times7}+15e^{-0.2\times2}$ and $13.754\ (mg)$ are required |

### Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $15e^{-0.2T} + 15e^{-0.2\times(T+5)} = 7.5$ | M1 | Attempts to write down a correct equation involving $T$ or $t$. Accept with or without correct bracketing, e.g. $15e^{-0.2T}+15e^{-0.2(T+5)}=7.5$ or $(15e^{-1}+15)e^{-0.2T}=7.5$ |
| $15e^{-0.2T} + 15e^{-0.2T}e^{-1} = 7.5$ | | |
| $15e^{-0.2T}(1+e^{-1}) = 7.5 \Rightarrow e^{-0.2T} = \dfrac{7.5}{15(1+e^{-1})}$ | dM1 | Attempts to solve their equation by proceeding to $e^{-0.2T}=\ldots$ An attempt should involve index law $x^{m+n}=x^m \times x^n$ and taking out a factor of $e^{-0.2T}$ |
| $T = -5\ln\!\left(\dfrac{7.5}{15(1+e^{-1})}\right)$ | A1 | Any correct form, e.g. $-5\ln\!\left(\dfrac{7.5}{15(1+e^{-1})}\right)$ |
| $= 5\ln\!\left(2 + \dfrac{2}{e}\right)$ | A1 | CSO $T = 5\ln\!\left(2+\dfrac{2}{e}\right)$. Condone $t$ appearing for $T$ throughout. |

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