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Ksenya-84 [330]
2 years ago
10

IM PUTTING THIS FOR 30 POINTS PLEASE ANSWER

Mathematics
1 answer:
Alex Ar [27]2 years ago
6 0

Answer:

Jacks scores 130 points

Jill scores 75 points

Step-by-step explanation:

J=Jacks score

K=Jills score

2K-20+K=205

So, we can do 3K-20=205

                            +20  +20

                                (cancel it out)

and we are left with 3K=225 which is K=75, so we do do 205-75=130(which is jacks score). To prove this we can do 75x2=150-20=130

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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
Have you ever thought about the number of times your heart beats in a life time? Consider the average life span of 75 years and
andreev551 [17]
Roughly 2,365,200,000. If you have muiltiple choices round up to the nearest
7 0
3 years ago
What were the lengths of the legs of a right triangle in which measures 19 degrees and the hypotenuses 15 units long
Otrada [13]

Answer:



The legs of a triangle with a hypotenuse that measures 15 units long have lengths that are equal to 15 times the sine or cosine of the given angle.

                     leg 1 = (15 units) x (cos 19) = 14.18 units

                     leg 2 = (15 units) x (sin 19) = 4.88 units

The lengths of the legs are 14.18 units and 4.88 units.  

8 0
3 years ago
(2³)² solve this problem
max2010maxim [7]

Answer:

(2^3)^2 = 64

Step-by-step explanation:

Option 1:

Using the following rule:

(a^n)^m = a^{nm}

Put in our expression,

a = 2

n = 3

m = 2

(a^n)^m = a^{nm}

(2^3)^2 = 2^{3*2}=2^6=64

Option 2:

Using the following rule:

a^n * a^m = a^{n+m}

Since our expression is the same as multiplying 2³ with itself, we can write it as a multiplication.

(2^3)^2 = 2^3 * 2^3

If we compare this with a^n * a^m, we can see that

a = 2

n = 3

m = 3 (in this case, n and m are equal)

a^n*a^m = a^{n+m}

2^3*2^3 = 2^{3+3} = 2^6 = 64

Answer: (2^3)^2 = 64

7 0
3 years ago
This week Aaron got a promotion at work that came with a 3 % pay increase. If now his monthly salary is $ 2163 , how much was he
NNADVOKAT [17]

Answer:

12.75

Step-by-step explanation:

5 0
3 years ago
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