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3 years ago

ASAP WHO EVER ANSWERS THIS WILL BE MARKED BRANLIEST

Mathematics

1 answer:

Kaylis [27]3 years ago

3 0

Answer:

option A is the answer hope you find it helpful

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Use green's theorem to compute the area inside the ellipse x252+y2172=1. use the fact that the area can be written as ∬ddxdy=12∫

Pavel [41]

The area of the ellipse $E$ is given by

$\displaystyle\iint_E\mathrm dA=\iint_E\mathrm dx\,\mathrm dy$

To use Green's theorem, which says

$\displaystyle\int_{\partial E}L\,\mathrm dx+M\,\mathrm dy=\iint_E\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)\,\mathrm dx\,\mathrm dy$

( $\partial E$ denotes the boundary of $E$ ), we want to find $M(x,y)$ and $L(x,y)$ such that

$\dfrac{\partial M}{\partial x}-\dfrac{\partial L}{\partial y}=1$

and then we would simply compute the line integral. As the hint suggests, we can pick

$\begin{cases}M(x,y)=\dfrac x2\\\\L(x,y)=-\dfrac y2\end{cases}\implies\begin{cases}\dfrac{\partial M}{\partial x}=\dfrac12\\\\\dfrac{\partial L}{\partial y}=-\dfrac12\end{cases}\implies\dfrac{\partial M}{\partial x}-\dfrac{\partial L}{\partial y}=1$

The line integral is then

$\displaystyle\frac12\int_{\partial E}-y\,\mathrm dx+x\,\mathrm dy$

We parameterize the boundary by

$\begin{cases}x(t)=5\cos t\\y(t)=17\sin t\end{cases}$

with $0\le t\le2\pi$ . Then the integral is

$\displaystyle\frac12\int_0^{2\pi}(-17\sin t(-5\sin t)+5\cos t(17\cos t))\,\mathrm dt$

$=\displaystyle\frac{85}2\int_0^{2\pi}\sin^2t+\cos^2t\,\mathrm dt=\frac{85}2\int_0^{2\pi}\mathrm dt=85\pi$

###

Notice that $x^{2/3}+y^{2/3}=4^{2/3}$ kind of resembles the equation for a circle with radius 4, $x^2+y^2=4^2$ . We can change coordinates to what you might call "pseudo-polar":

$\begin{cases}x(t)=4\cos^3t\\y(t)=4\sin^3t\end{cases}$

which gives

$x(t)^{2/3}+y(t)^{2/3}=(4\cos^3t)^{2/3}+(4\sin^3t)^{2/3}=4^{2/3}(\cos^2t+\sin^2t)=4^{2/3}$

as needed. Then with $0\le t\le2\pi$ , we compute the area via Green's theorem using the same setup as before:

$\displaystyle\iint_E\mathrm dx\,\mathrm dy=\frac12\int_0^{2\pi}(-4\sin^3t(12\cos^2t(-\sin t))+4\cos^3t(12\sin^2t\cos t))\,\mathrm dt$

$=\displaystyle24\int_0^{2\pi}(\sin^4t\cos^2t+\cos^4t\sin^2t)\,\mathrm dt$

$=\displaystyle24\int_0^{2\pi}\sin^2t\cos^2t\,\mathrm dt$

$=\displaystyle6\int_0^{2\pi}(1-\cos2t)(1+\cos2t)\,\mathrm dt$

$=\displaystyle6\int_0^{2\pi}(1-\cos^22t)\,\mathrm dt$

$=\displaystyle3\int_0^{2\pi}(1-\cos4t)\,\mathrm dt=6\pi$

3 0

3 years ago

Luis wants to buy a skateboard that usually sells for $79.47. All merchandise is discounted by 12%. What is the total cost of th

valentinak56 [21]

79.47 (total cost before sale) - 9.02 (12% off) = 70.45 (total after sale).
70.45 (total after sale) + 6.32 (tax) = $76.77
Hope this helps!!!

7 0

3 years ago

Tia’s car needs repairs. Honest Harry will charge $70 per hour plus $130 for the part. Lucky Lou will charge $80 per hour plus $

elena-s [515]

To answer the question above, let x the number of hours for the two jobs to cost equal. Harry's charge can be expressed as 70x + 130 and that of Lou is 80x + 40. Equating both,

70x + 130 = 80x + 40 ; x = 9

Thus, the job should take 9 hours for both charges to be equal.

3 0

3 years ago

Read 2 more answers

The ratio of grade 7's to grade 8's is 5:3. If there

Greeley [361]

Answer:

240 students

Step-by-step explanation:

Grade 7 : Grade 8 = 5 : 3

Number of seventh grade students = 5x

Number of Eighth grade students = 3x

3x = 90

x = 90÷3

x = 30

Total students = 5x + 3x = 8x = 8 * 30 = 240

5 0

3 years ago

Circle R has equation

Rudiy27

Answer:

The center is (-10,10) and the radius is 4sqrt(3)

Step-by-step explanation:

(x + 10)^2 + (y - 10)^2 = 48

We can write the equation of a circle as

(x -h)^2 + (y - k)^2 = r^2 where (h,k) is the center and r is the radius

(x- -10)^2 + (y - 10)^2 = (sqrt(16*3) )^2

(x- -10)^2 + (y - 10)^2 = (4sqrt(3)) ^2

The center is (-10,10) and the radius is 4sqrt(3)

5 0

3 years ago