Solution — 2017 Question 5 Graphs| Simultaneous Linear Equations

(a)

Let $r,$ $s$ and $w$ be the cost (in dollars) of a pair of Runners, Supers and Walkers respectively.

$\begin{matrix} s & = 2 w \\ s - 2 w & = 0 & (1) \end{matrix}$

[Supers cost twice as much as Walkers]

$\begin{matrix} 10 r & = 7 s + 50 \\ - 7 s + 10 r & = 50 & (2) \end{matrix}$

[10 pairs of Runners cost 50 more than 7 pairs of Supers]

$\begin{matrix} 2 s + 6 r + 4 w & = 481 & (3) \end{matrix}$

[total cost of 2 Supers, 6 Runners and 4 Walkers]

Using a GC to solve the equations,

$\begin{matrix} r = 43.50, s = 55, w = 27.50 \end{matrix}$

[solution to the system]

$\begin{matrix} Cost of Runners & = $ 43.50 ∎ \end{matrix}$

[required answer]

(b)

$\begin{matrix} Total cost & = 100 (55 + \frac{87}{2} + \frac{55}{2}) \\ = 12 600 \end{matrix}$

[cost price of manufacturing]

$\begin{matrix} Total revenue & = 100 (3 (80)) \\ = 24 000 \end{matrix}$

[selling price of shoes]

$\begin{matrix} Profit & = 24 000 - 12 600 \\ = $ 11 400 ∎ \end{matrix}$

(c)

(d)

From GC,

$\begin{matrix} Maximum value of P & = 13.61 ∎ \end{matrix}$

$\begin{matrix} Value of x & = 15.12 ∎ \end{matrix}$

(e)

When $x = 55,$

$\begin{matrix} P & = 7 \sqrt{55} - 0.9 (55) \\ = 2.4134 \end{matrix}$

[find profit when x=55]

$\begin{matrix} Selling price & = 55 + 2.4134 \\ = 57.413 \\ = 57.41 ∎ (to 2 d.p.) \end{matrix}$

[find selling price]

(f)

When $x = 65,$

$\begin{matrix} P & = 7 \sqrt{65} - 0.9 (65) \\ = - 2.0642 \end{matrix}$

[find profit when x=65]

Since $P = - 2.0642 < 0,$ the economist predicts a negative profit (i.e. loss) when the manufacturing cost is $$ 65.$ Hence we should advise the company to not product Extremes so as not to make a loss $∎$