9 A market trader notices that daily sales are dependent on two variables:
number of hours, \(t\), after the stall opens
total sales, \(x\), in pounds since the stall opened.
The trader models the rate of sales as directly proportional to \(\frac { 8 - t } { x }\)
After two hours the rate of sales is \(\pounds 72\) per hour and total sales are \(\pounds 336\)
9
- Show that
$$x \frac { \mathrm {~d} x } { \mathrm {~d} t } = 4032 ( 8 - t )$$
9
- Hence, show that
$$x ^ { 2 } = 4032 t ( 16 - t )$$
\(\mathbf { 9 }\) (c) The stall opens at 09.30.
9
- The trader closes the stall when the rate of sales falls below \(\pounds 24\) per hour.
Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall.
9
- (ii) Explain why the model used by the trader is not valid at 09.30.