Tsunami Arrival Times, Numerical Integration

J

Jaskaran

Junior Member
Joined
May 5, 2006
Messages
67
Hey all, I am a bit stumped on what this problem is asking me, any amount of help would be appreciated!

Scientists estimate the arrival times of tsunamis based on the point of origin P and ocean depths. The speed \(\displaystyle s\) of a tsunami in miles per hour is approximately \(\displaystyle s= \sqrt{15d}\), where \(\displaystyle d\) is the ocean depth in feet.

(a) let f(x) be the ocean depth x miles from P (in the direction of the coast). Argue using Riemann sums that the time T required for the tsunami to travel M miles toward the coast is \(\displaystyle T=\int_{0}^{M}\frac{dx}{\sqrt{15f(x)}}\).

(b) Use Simpsons rule to estimate T if M = 1,000 and the ocean dpeth (in feet), measured at 100-mile intervals starting from P, are {0, 13000}, {100, 11500}, {200, 10500}, {300, 9000}, {400, 8500}, {500, 7000}, {600, 6000}, {700, 4400}, {800, 3800}, {900, 3200}, {1000, 2000}
Click to expand...


What do they mean "argue" in part a? That I should strictly use Simpsons rule? Or other types of Riemann sums? (Trapezoid, Midpoint)

ANY help is appreciated!
 
I think you have a typo. I am pretty sure It should be

\(\displaystyle \int_{0}^{M}\frac{1}{\sqrt{15f(x)}}dx\)
 
Jaskaran said:
Hey all, I am a bit stumped on what this problem is asking me, any amount of help would be appreciated!

Scientists estimate the arrival times of tsunamis based on the point of origin P and ocean depths. The speed \(\displaystyle s\) of a tsunami in miles per hour is approximately \(\displaystyle s= \sqrt{15d}\), where \(\displaystyle d\) is the ocean depth in feet.

(a) let f(x) be the ocean depth x miles from P (in the direction of the coast). Argue using Riemann sums that the time T required for the tsunami to travel M miles toward the coast is \(\displaystyle T=\int_{0}^{M}\frac{dx}{\sqrt{f(x)}}\).

(b) Use Simpsons rule to estimate T if M = 10,000 and the ocean dpeth (in feet), measured at 100-mile intervals starting from P, are {0, 13000}, {100, 11500}, {200, 10500}, {300, 9000}, {400, 8500}, {500, 7000}, {600, 6000}, {700, 4400}, {800, 3800}, {900, 3200}, {1000, 2000}
Click to expand...


What do they mean "argue" in part a? That I should strictly use Simpsons rule? Or other types of Riemann sums? (Trapezoid, Midpoint)

ANY help is appreciated!
Click to expand...

In the given context, "argue" is synonymous to "prove".
 
Think about d=rt.

The rate is given. They tell you that x is the distance.

Using Simpson's rule and the given formula, break it up into 10 subintervals from 0 to 1000 and add them up.

f(x) is the y values in part b you use in Simpson's rule.
 
galactus said:
I think you have a typo. I am pretty sure It should be

\(\displaystyle \int_{0}^{M}\frac{1}{\sqrt{15f(x)}}dx\)
Click to expand...
Heh, you're right, forgot the 15...
 
Yes, I know.

Do you see what SK and I are getting at?.

x=distance, s=rate=\(\displaystyle \sqrt{15f(x)}\)

velocity is the derivative of the position function.

\(\displaystyle \frac{dx}{dt}=\sqrt{15f(x)}\)

\(\displaystyle dx=\sqrt{15f(x)}dt\)

\(\displaystyle \frac{dx}{\sqrt{15f(x)}}=dt\)

Integrate both sides:

\(\displaystyle t=\int\frac{dx}{\sqrt{15f(x)}}\)

You can think of it as x=st. Where x is the distance and x/t=s. A small change in distance is dx and a small change in time is dt.

If you make a diagram of the Riemann sum and draw some small rectangles under the curve, you can see that the width of the

rectangle is dx. The area under the curve is t with limits from 0 to M.

To use Simpson, you know that f(x) is the ocean depth x miles from shore. Plug in the given values into Simpson's formula and add them up.

\(\displaystyle \int_{a}^{b}f(x)=\left(\frac{b-a}{3n}\right)\left[y_{0}+4y^{1}+2y_{2}+4y_{3}+2y_{4}+\cdot\cdot\cdot\cdot 2y_{n-2}+4y_{n-1}+y_{n}\right]\)

f(x) in the formula above is your \(\displaystyle \frac{1}{\sqrt{15f(x)}}\). In this case, M=1000.
 
You must log in or register to reply here.
Top