The length of the parametric curve (call it <em>C </em>) is given by
![\displaystyle\int_C\mathrm ds=\int_0^{8\pi}\sqrt{\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+\left(\frac{\mathrm dy}{\mathrm dt}\right)^2}\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_C%5Cmathrm%20ds%3D%5Cint_0%5E%7B8%5Cpi%7D%5Csqrt%7B%5Cleft%28%5Cfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%5Cright%29%5E2%2B%5Cleft%28%5Cfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%5Cright%29%5E2%7D%5C%2C%5Cmathrm%20dt)
We have
![x=t-2\sin t\implies\dfrac{\mathrm dx}{\mathrm dt}=1-2\cos t](https://tex.z-dn.net/?f=x%3Dt-2%5Csin%20t%5Cimplies%5Cdfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%3D1-2%5Ccos%20t)
![y=1-2\cos t\implies\dfrac{\mathrm dy}{\mathrm dt}=2\sin t](https://tex.z-dn.net/?f=y%3D1-2%5Ccos%20t%5Cimplies%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%3D2%5Csin%20t)
Now,
![\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+\left(\frac{\mathrm dy}{\mathrm dt}\right)^2=5-4\cos t](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%5Cright%29%5E2%2B%5Cleft%28%5Cfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%5Cright%29%5E2%3D5-4%5Ccos%20t)
so that the arc length integral reduces to
![\displaystyle\int_0^{8\pi}\sqrt{5-4\cos t}\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_0%5E%7B8%5Cpi%7D%5Csqrt%7B5-4%5Ccos%20t%7D%5C%2C%5Cmathrm%20dt)
which has an approximate value of 53.4596.
<h2>Hey there!</h2><h2>
![\bold{Distribute\downarrow}\\ \bold{2x(12) + 2x(4) -49}\\ \bold{2x(12)=24x}\\ \bold{2x(4) =8x}\\ \bold{24x+8x-49}\\\\\bold{Combine\ your\ like\ terms \downarrow}\\\bold{(24x + 8) + (-49)}\\\bold{24x+8=32x}\\\bold{-49 =-49}\\\\\\ \\ \boxed{\bold{Possible\ answer\ \#1: 32x - 49}}\checkmark](https://tex.z-dn.net/?f=%5Cbold%7BDistribute%5Cdownarrow%7D%5C%5C%20%5Cbold%7B2x%2812%29%20%2B%202x%284%29%20-49%7D%5C%5C%20%5Cbold%7B2x%2812%29%3D24x%7D%5C%5C%20%5Cbold%7B2x%284%29%20%3D8x%7D%5C%5C%20%5Cbold%7B24x%2B8x-49%7D%5C%5C%5C%5C%5Cbold%7BCombine%5C%20your%5C%20like%5C%20terms%20%5Cdownarrow%7D%5C%5C%5Cbold%7B%2824x%20%2B%208%29%20%2B%20%28-49%29%7D%5C%5C%5Cbold%7B24x%2B8%3D32x%7D%5C%5C%5Cbold%7B-49%20%3D-49%7D%5C%5C%5C%5C%5C%5C%20%5C%5C%20%5Cboxed%7B%5Cbold%7BPossible%5C%20answer%5C%20%5C%231%3A%2032x%20-%2049%7D%7D%5Ccheckmark)
</h2><h2 /><h2>
![\bold{Do\ PEMDAS\ to\ solve\downarrow}\\\bold{2(12+4) -49}\\\bold{12+4=16}\\\bold{2(16)-49}\\\bold{2(16)=32}\\\bold{32-49=-17}\\ \\ \\ \\ \\ \boxed{\bold{Possible\ answer\ \#2: -17}}}\checkmark](https://tex.z-dn.net/?f=%5Cbold%7BDo%5C%20PEMDAS%5C%20to%5C%20solve%5Cdownarrow%7D%5C%5C%5Cbold%7B2%2812%2B4%29%20-49%7D%5C%5C%5Cbold%7B12%2B4%3D16%7D%5C%5C%5Cbold%7B2%2816%29-49%7D%5C%5C%5Cbold%7B2%2816%29%3D32%7D%5C%5C%5Cbold%7B32-49%3D-17%7D%5C%5C%20%5C%5C%20%5C%5C%20%5C%5C%20%5C%5C%20%5Cboxed%7B%5Cbold%7BPossible%5C%20answer%5C%20%5C%232%3A%20-17%7D%7D%7D%5Ccheckmark)
</h2><h2>
![\bold{Note\downarrow}\\\bold{PEMDAS\ means......}\\\\\bold{Parentheses}\\\bold{Exponents} \\\bold{Multiplication}\\\bold{Division} \\\bold{Addition}\\\bold{Subtraction}](https://tex.z-dn.net/?f=%5Cbold%7BNote%5Cdownarrow%7D%5C%5C%5Cbold%7BPEMDAS%5C%20means......%7D%5C%5C%5C%5C%5Cbold%7BParentheses%7D%5C%5C%5Cbold%7BExponents%7D%20%5C%5C%5Cbold%7BMultiplication%7D%5C%5C%5Cbold%7BDivision%7D%20%5C%5C%5Cbold%7BAddition%7D%5C%5C%5Cbold%7BSubtraction%7D)
</h2><h2>Good luck on your assignment and enjoy your day! </h2><h3>~
![\frak{LoveYourselfFirst:)}](https://tex.z-dn.net/?f=%5Cfrak%7BLoveYourselfFirst%3A%29%7D)
</h3>
The fraction of the cake that was ron's pieces is 1/5.
<h3>What is the fraction of the cake was ron's pieces?</h3>
Ratio demonstrates how many times one number can fit into another number. Ratios contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).
Since Mrs Jones gave 1/5 of a cake to her neighbour, the remaining fraction will be:
= 1 - 1/5
= 4/5
The fraction of the cake that was ron's pieces will be:
= 2/(1+3+4) × 4/5
= 2/8 × 4/5
= 8/40
= 1/5
Learn more about ratio on:
brainly.com/question/2328454
#SPJ1
9a-ab+5b
9(2)-(2)(7)+5(7)
18-14+35
39
Answer:
6.5 x 10^-7
Step-by-step explanation: