Question

picture enclosed. To monitor the thermal pollution of a river, a biologist takes hourly temperature reading (in F) from 9 am to 5 pm. the results are shown in the following table: Use simpsons rule and the definition on page 452 for average value to estimate the average water temperature between 9am and 5pm

Answer 1

pg. 452

Answer 2

Alright well we have Simpson's method for numerical approximation like so:
\[\int^b_af(x)\phantom{.}dx\approx.\frac{b-a}{6}\left[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)\right]\]
Also, if we have a function \(f(x)\), defined on the interval \([a,b]\), then we also have the average value of the function in the interval as such:
\[AVERAGE\phantom{.}\{f_{[a,b]}\}=\frac{\int^b_af(x)\phantom{.}dx}{b-a}\]
So then we can simplify substituting Simpson's method instead of the integral:
\[\eqalign{
AVERAGE\phantom{.}\{f_{[a,b]}\}&\approx\frac{\frac{b-a}{6}\left[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)\right]}{b-a} \\
&\approx\frac{f(a)+4f\left(\frac{a+b}{2}\right)+f(b)}{6} \\
}\]

Answer 3

So therefore, we need the beginning x-interval \((a)\) and the end x-interval \((b)\).

Let us define the function \(f(x)\) as one where x represents the time in hours after 9AM and \(f\) defines the water temperature in degrees \((^\circ F)\)Looking at the data, we can tell that the interval is \(x=[0,8]\) where \(x=0\) would be zero hours after 9AM (9:00AM) and \(x=8\) would be 8 hours after 9AM (17:00=5:00PM). So therefore we can set up our data:
\[AVERAGE\phantom{.}\{f_{[0,8]}\}\approx\frac{f(0)+4f\left(\frac{0+8}{2}\right)+f(8)}{6}\]
Now we can simplify:
\[AVERAGE\phantom{.}\{f_{[0,8]}\}\approx\frac{f(0)+4f(4)+f(8)}{6}\]
Now, analyzing the data, we can tell that:
\[AVERAGE\phantom{.}\{f_{[0,8]}\}\approx\frac{75.3+4(86.5)+75.1}{6}\]
And we can simplify once more:
\[AVERAGE\phantom{.}\{f_{[0,8]}\}\approx\frac{75.3+4(86.5)+75.1}{6}\approx82.7^\circ F\]
And there you go!