DP Mathematics: Applications and Interpretation Questionbank

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

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 $d$.

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

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 $d=25$.

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 $k$.

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

find the value of $a$ and of $b$.

the value of $m$ when $p=0$.

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

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

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 $n$. 

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

predict the temperature of the metal rod after 3 minutes.

(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 $\frac{1}{T}$.

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

Find the value of this area.

Write down the coefficient of determination.

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.

Show that $\text{ln}y=qx+\text{ln}p$.

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

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

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

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

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