$[$Maximum mark: $17]$

The time taken for a computer to process a task is defined by the number of operations it needs to perform. The number of operations, $P,$ and the time taken, $t$ seconds, for a specific algorithm were recorded. This information is summarized as follows:

When $p=100,t=2s$;

When $p=500,t=10s$;

When $p=1000,t=40$ :

This information was used to create Model C $,$ where $t$ is a function of $p.$

$t(p)=rp2+sp,r,sinZ$

At $100$ operations, Model $C$ can be represented by the equation.

$100r+s=2$

Write down a second equation to represent Model $C,$ when the number of operations is $500.$

Find $r$ and $s.$

Find the coordinates of the vertex of the graph of $y=t(p)$

Using the values given and your answer to part $($b$),$ sketch the graph of $y=t(p)$ for $0p1000$ and $0t50,$ clearly showing the vertex.

Hence, identify why Model C may not be appropriate for very large numbers of operations.

$[1]$

Additional data was used to create Model D$,$ a revised model for the time taken by the algorithm.

Model $D$: $t(p)=0\mathrm{.}8p2-2\mathrm{.}5p$

Use Model D to calculate an estimate for the time taken for $2000$ operations. $[2]$

$7.$ The actual time taken for $2000$ operations is $3200$ seconds. Calculate the percentage error in the estimate in part $(f).$

$[2]$

$8.$ It is found that once an algorithm starts, there is an average delay of $0\mathrm{.}5$ seconds before the first operation is executed. An algorithm is given a task with o operations and must complete within $5000$ seconds.

Using Model D and taking the delay into account, calculate the maximum number of operations the algorithm can handle.

$[3]$