With reference to the shape of the graph, explain whether your answer to part (a)(i) will be an over-estimate or an underestimate of the area.

Write down the coefficient of determination.

(i)   the gradient of this line in terms of $c$;

(ii)  the $y$-intercept of this line in terms of $A$.

Find the equation of the regression line for $\ln k$ on $\frac{1}{T}$.

Find the value of this area.

Hence explain how a straight line graph could be drawn using the coordinates in the table.

Show that $\text{ln}\left(T-25\right)=bt+\text{ln}a$.

The rate, $A$, of a chemical reaction at a fixed temperature is related to the concentration of two compounds, $B$ and $C$, by the equation

$A=k{B}^x{C}^y$, where $x$, $y$, $k\in \mathbb{R}$.

A scientist measures the three variables three times during the reaction and obtains the following values.

Find $x$, $y$ and $k$.

It is believed that two variables, $v$ and $w$ are related by the equation $v=k{w}^n$, where $k\text{,}n\in \mathbb{R}$.  Experimental values of $v$ and $w$ are obtained. A graph of $\text{ln}v$ against $\text{ln}w$ shows a straight line passing through (−1.7, 4.3) and (7.1, 17.5).

Find the value of $k$ and of $n$. 

predict the temperature of the metal rod after 3 minutes.

Use the trapezoidal rule to find an estimate for the area.

Use all the coordinates in the table to find the equation of the least squares cubic regression curve.

a formula for $m$ in terms of $p$.

Show that $\text{ln}y=qx+\text{ln}p$.

Explain why this graph indicates that $P$ is inversely proportional to $d^n$.

Use your answer to part (b) to write down the value of $n$ to the nearest integer.

Find the equation of the regression line of $\text{ln}\left(T-25\right)$ on $t$.

By finding the equation of a suitable regression line, show that $p=1.83$ and $q=0.986$.

Hence find the area enclosed by the exponential function, the $x$-axis, the $y$-axis and the line $x=4.4$.

Find the equation of the straight line, giving your answer in the form $\text{ln}m=ap+b$, where $a\text{,}b\in \mathbb{R}$.

Write down an expression for the area enclosed by the cubic function, the $x$-axis, the $y$-axis and the line $x=4.4$.

find the value of $a$ and of $b$.

Find the equation of the least squares regression line of ${\log }_{10} P$ against ${\log }_{10} d$.

Find an expression for $P$ in terms of $d$.

Use the data in the second table to find the value of $m$ and the value of $b$ for the regression line, $\ln x=m(\ln d)+b$.

Assuming that the model found in part (a) remains valid, estimate the percentage of trees in stock when $d=25$.