MHB Compute Pool Volume: Angeezzzz's Question at Yahoo Answers

  • Thread starter Thread starter MarkFL
  • Start date Start date
  • Tags Tags
    Computing Volume
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Volume integration help?


As viewed from above, a swimming pool has the shape of the ellipse given by
x^2/3600+y^2/2500=1

The cross sections perpendicular to the ground and parallel to the y-axis are squares. Find the total volume of the pool. (Assume the units of length and area are feet and square feet respectively. Do not put units in your answer.)

V= ? ft^3

I have posted a link there to this thread so the OP can view my work.
 
Mathematics news on Phys.org
Hello Angeezzzz,

We first should write the ellipse in standard form:

$$\frac{x^2}{60^2}+\frac{y^2}{50^2}=1$$

Thus, we see the length of the semi-major axis is 60. We may restrict ourselft to the first quadrant, and then by symmetry quadruple the result to get the total volume. The volume of an arbitrary rectangular slice is:

$$dV=bh\,dx$$

where:

$$b=y=\frac{\sqrt{3000^2-50^2x^2}}{60}$$

$$h=2y=\frac{\sqrt{3000^2-50^2x^2}}{30}$$

Hence, we may state:

$$dV=\frac{3000^2-50^2x^2}{1800}\,dx=-\frac{25}{18}\left(x^2-3600 \right)\,dx$$

And so the total volume is given by:

$$V=-\frac{50}{9}\int_0^{60}x^2-3600\,dx$$

Applying the FTOC, we obtain:

$$V=-\frac{50}{9}\left[\frac{x^3}{3}-3600x \right]_0^{60}=-\frac{50\cdot60^3}{9}\left(\frac{1}{3}-1 \right)=-\frac{50\cdot60^3}{9}\left(-\frac{2}{3} \right)=800000$$
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top