Solution — 2024 Question 5 Exponential & Logarithmic Functions| Simultaneous Linear Equations| Differentiation & Applications| Integration & Applications

(a)

Let $b,$ $c$ and $d$ denote the number of bedside, console and dining tables produced each month respectively.

\begin{matrix} b & = 3 (c + d) \\ b - 3 c - 3 d & = 0 & (1) \end{matrix}

bedside is 3 times console and dining combined

\begin{matrix} d & = c + 29 \\ c - d & = - 29 & (2) \end{matrix}

29 more dining than console

\begin{matrix} b + c + d & = 156 & (3) \end{matrix}

total tables

Solving equations (1), (2) and (3) with a GC,

\begin{matrix} b = 117, c = 5, d = 34 \end{matrix}

solution to the system of equations

The factory produces 117 bedside tables, 5 console tables and 34 dining tables each month. $∎$

(b)

When $t = 5.5,$

\begin{matrix} S & = 600 (1.2 - 0.3 e^{1 - 0.1 t}) \\ = 600 (1.2 - 0.3 e^{1 - 0.1 (5.5)}) \\ = 600 (1.2 - 0.3 e^{0.450 00}) \\ = 437.70 \\ = 438 ∎ (to 3 s.f.) \end{matrix}

(c)

When $S = 500,$

\begin{matrix} 600 (1.2 - 0.3 e^{1 - 0.1 t}) & = 500 \end{matrix}

Dividing by $600$ and rearranging,

\begin{matrix} \frac{3}{10} e^{1 - 0.1 t} & = \frac{11}{30} \\ e^{1 - 0.1 t} & = \frac{11}{9} \\ \ln e^{1 - 0.1 t} & = \ln \frac{11}{9} \\ 1 - 0.1 t & = \ln \frac{11}{9} \\ 0.1 t & = - \ln \frac{11}{9} + 1 \\ t & = - 10 \ln \frac{11}{9} + 10 \\ = 7.9933 \\ = 7.99 ∎ (to 3 s.f.) \end{matrix}

(d)

\begin{matrix} S & = 600 (1.2 - 0.3 e^{1 - 0.1 t}) \\ \frac{d S}{d t} & = 600 (0.03 e^{1 - 0.1 t}) \\ = 18 e^{1 - 0.1 t} ∎ \end{matrix}

Since $e^{1 - 0.1 t} > 0$ for all $t,$ we have $\frac{d S}{d t} > 0$ for all $t \geq 0.$

This means that $S$ is always increasing for $t \geq 0,$ i.e. the model indicates that the rate of ${CO}_{2}$ emissions will always be increasing over time. $∎$

(e)

\begin{matrix} \int_{0}^{3} 600 (1.2 - 0.3 e^{1 - 0.1 t}) d t \\ = 600 \int_{0}^{3} (1.2 - 0.3 e^{1 - 0.1 t}) d t \\ = 600 {[1.2 t + 3 e^{1 - 0.1 t}]}_{0}^{3} \\ = 600 (1.2 (3) + 3 e^{1 - 0.1 (3)} - (1.2 (0) + 3 e^{1 - 0.1 (0)})) \\ = 892 ∎ (to 3 s.f.) \end{matrix}

It represents the total ${CO}_{2}$ emissions (in tonnes) over the first 3 years.