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 $\frac{1}{T}$.

the value of $m$ when $p=0$.

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.

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 $d=25$.