Question 201400
In general when you want to find the equation of a line, you need the slope of the line and at least one point on the line. To start with you have two points:
<pre>
Year   t (Year-1920)   R(t)   Point
1920   0               46.4   (0, 46.4)
1990   70              45.0   (70, 45.0)
</pre>
We have the point(s) we need but we do not have the slope. With the two points we can calculate the slope using the slope formula:
{{{m = (y[2]-y[1])/(x[2]-x[1])}}}
Substituting the coordinates of the two points above in the formula:
{{{m = (46.4 - 45.0)/(0 - 70)}}}
Subtracting we get
{{{m = (1.4)/(-70)}}}
Dividing 1.4 by -70 we get
{{{m = -0.02}}}
Now we have the slope and at least one point. There are a couple of ways to find the equation of the line:<ul><li>Using the Point-Slope formula: {{{y - y[1] = m(x - x[1])}}}</li><li>Using the Slope-Intercept form: {{{y = mx + b}}}</li></ul>Since most students seem to the prefer the second method that is how we will proceed.
First we substitute the slope and the coordinates of a point (it doesn't matter which point as long as it is on the line) into the Slope-Intercept form.
{{{y = mx + b}}}
Substituting the coordinates of (0, 46.4) in for the x and the y and substituting -0.02 in for the m we get
{{{(46.4) = (-0.02)*(0) + b}}}
We can now solve for b. Since -0.02 * 0 is zero and since 0 + b is b we get
{{{46.4 = b}}}
Now that we have m (-0.02) and b (46.4) we can write the equation of our line. In doing so we will use t instead of x and R(t) instead of y.
{{{R(t) = -0.02t + 46.4}}}
which is the answer to part (a).<br>
For part (b), to calculate the record in 2003 we will use 83 for t (since 2003-1920 = 83) and figure out R(83). To calculate the record in 2006 we will use t = 86 and calculate R(86).
{{{R(83) = -0.02(83) + 46.4}}}
Multiplying -0.02 by 83 we get
{{{R(83) = -1.66 + 46.4}}}
Adding we get
{{{R(83) = 44.74}}}
Similarly
{{{R(86) = -0.02(86) + 46.4}}}
Simplifying like above
{{{R(86) = 44.68}}}<br>
For part (c) we will use R(t) = 44.56 and solve for t:
{{{44.56 = -0.02t + 46.4}}}
Subtracting 46.4 from both sides we get
{{{-1.84 = -0.02t}}}
Dividing both sides by -0.02 we get
{{{92 = t}}}
Since t = 92 the year must be 92+1920 or 2012