Question 13 - A-Level Maths

13.

Prove that for all positive values of $x$ and $y$ $$\sqrt { x y } \leqslant \frac { x + y } { 2 }$$
Prove by counter example that this is not true when $x$ and $y$ are both negative. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{106e8e6b-912d-45ef-89e9-12ffc04bfd25-22_844_1427_278_406} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A town's population, $P$, is modelled by the equation $P = a b ^ { t }$, where $a$ and $b$ are constants and $t$ is the number of years since the population was first recorded. The line $l$ shown in Figure 2 illustrates the linear relationship between $t$ and $\log _ { 10 } P$ for the population over a period of 100 years.
The line $l$ meets the vertical axis at $( 0,5 )$ as shown. The gradient of $l$ is $\frac { 1 } { 200 }$.
Write down an equation for $l$.
Find the value of $a$ and the value of $b$.
With reference to the model interpret
1. the value of the constant $a$,
2. the value of the constant $b$.
Find
1. the population predicted by the model when $t = 100$, giving your answer to the nearest hundred thousand,
2. the number of years it takes the population to reach 200000 , according to the model.
State two reasons why this may not be a realistic population model.
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