MHB Find the areas of segment in circle

  • Thread starter Thread starter Etrujillo
  • Start date Start date
  • Tags Tags
    Areas Circle
Etrujillo
Messages
9
Reaction score
0
So far i have.
14) area of sector is πr²/3 = 12π
length of chord. that triangle has two sides of 6 and angle of 120º
split the triangle in two right triangles with angle of 120/2 = 60 and hyp=6. other (longer) side is:
sin 60 = x/6
s = 6 sin 60 = 6(√3/2) = 3√3
third side is
s = 3 cos 60 = 3
area =(1/2)(3)(3√3) = 4.5√3, double for both triangles
subtract that from the sector to get (12π) – (9√3)

15) similar to above.
find the area of the 270º sector and add the area of the triangle
area of sector is (270/360)(π9²) or (3/4)81π
area of triangle is (1/2)81

Is this correct?
What can i do differently?

View attachment 8705
 

Attachments

  • 20181204_093323-3.jpg
    20181204_093323-3.jpg
    14.1 KB · Views: 116
Mathematics news on Phys.org
Re: Find the area of the shared region.

14.) I would take the area of the circular sector, and subtract from it the area of the triangle to get the shaded area \(A\):

$$A=\frac{1}{2}r^2\theta-\frac{1}{2}r^2\sin(\theta)=\frac{r^2}{2}(\theta-\sin(\theta))$$

Next, we identify:

$$r=6\text{ cm}$$

$$\theta=\frac{2\pi}{3}$$

Hence:

$$A=\frac{(6\text{ cm})^2}{2}\left(\frac{2\pi}{3}-\sin\left(\frac{2\pi}{3}\right)\right)=3\left(4\pi-3\sqrt{3}\right)\text{ cm}^2\quad\checkmark$$

This is equivalent to the area you stated. :D

15.) I would find the sum of 3/4 of the area of the circle and the right isosceles triangle:

$$A=\frac{3}{4}\pi r^2+\frac{1}{2}r^2=\frac{r^2}{4}(3\pi+2)$$

We identify:

$$r=9\text{ in}$$

And so:

$$A=\frac{(9\text{ in})^2}{4}(3\pi+2)=\frac{81}{4}(3\pi+2)\text{ in}^2\quad\checkmark$$

This is equivalent to what you would get when you add the two areas you found.
 
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