Problem:
A boat drives a steady course with a variable speed for 4 hours. The speed is registered at regular intervals in meters per second.
The registration shows $2.4, 4.4, 7.6, 8.4, 8.6, 7.9, 8.3, 8.7, 7.7, 6.5, 7.1, 6.7, 1.4$ (sorry, but I'm making a point out of this later).
Use Simpson's Rule to estimate how far the boat has traveled during the four hours, and its average speed.
Additional question; why can Simpson's Rule be used to estimate the boat's distance traveled?
My progress:
There are 13 recordings, which is an odd number, so that should be fine.
Using the Composite Simpson's Rule (with coefficients 1,4,2,4,2…,2,4,1) , I get $$\frac19\left[ 2.4 + 4(4.4) + 2(7.6) + 4(8.4) + \ldots + 4(6.7) + 1.4 \right] \approx 28.09$$ (unless I've made some careless mistake).
Here $\frac19$ comes from $\frac h3$ (from formula) where $h = \frac13$.
Now, as I understand it, this number would – since we're estimating the area under the "curve" (i.e. using Simpson to estimate a definite integral given points) – be a number for the distance traveled.
But since the numbers are given in $\frac ms$, wouldn't this $28.09m$ be an oddly low number? What am I missing here?
Any help appreciated!
Best Answer
I believe your $\frac{1}{9}$ is obtained by $\frac{1}{3}\cdot\frac{1}{3}$ where one of the $\frac{1}{3}$ is in hours. So instead you should multiply by $3600$, which is the number of seconds in an hour.