Review question

Ref: R8775

## Solution

 $x\vphantom{4}$ $\smash{y}\vphantom{4}$ $4$ $7$ $10$ $13$ $96$ $133$ $160$ $117$

In the above table the values of $y$ were calculated from a formula $y = Ax^2 + Bx$, where $A$ and $B$ are constants. One of the values of $y$, however, has been deliberately misprinted.

Plot the values of $y/x$ against those of $x$

We have the following calculated data:

 $x\vphantom{4}$ $\smash{y/x}\vphantom{4}$ $4$ $7$ $10$ $13$ $24$ $19$ $16$ $9$

The data are plotted below.

1. find which value of $y$ has been misprinted and give the correct value,

If $y = Ax^2 + Bx$, then $y/x = Ax + B$.

This is the equation of a straight line, so, given the information in the question, we should expect to find a line through three of the four points plotted above. This is in fact the case!

From the plot above, the incorrect value of $y$ is the one corresponding to $x=10$, namely $160$.

Since the correct value of $y/x$ must lie on the line, this value is $14$, and so the correct value of $y$ is $14\times10=140$.

1. estimate the numerical values of $A$ and $B$.

The equation of the blue line is $y/x = Ax + B$, so we can find the value of $A$ by finding the gradient of the line and the value of $B$ by finding its $y$-intercept.

The gradient of the line in the plot above is (from looking at the points where $x=4$ and $x=13$) $\frac{\Delta y}{\Delta x} = \frac{9 - 24}{13 - 4} = -\frac{15}{9} = -\frac{5}{3}$ so $A = -\dfrac{5}{3}$. Taking the point with $x=4$, $y/x=24$, we have $24=-\frac{5}{3}\times 4+B$ so $B=\dfrac{92}{3}=30\frac{2}{3}$.

If we were doing this problem using real-world data rather than calculated figures, we would not use our data points to find the equation of the line (and hence $A$ and $B$, but would instead use points on the line itself. In this case, though, these are the same thing.