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3 years ago

What is the maximum number of possible solutions for the system shown below?

1 answer:

7 0

$\bf \begin{cases} 3x^2+y^2=4\\[-0.5em] \hrulefill\\ x^2+y=5\implies \boxed{y}=5-x^2 \end{cases} \\\\\\ \stackrel{\textit{substituting \boxed{y} in the 1st equation}}{3x^2+\left( \boxed{5-x^2} \right)^2=4}\implies 3x^2+25-10x^2+x^4=4 \\\\\\ x^4-8x^2+21=0\impliedby \begin{array}{llll} \textit{the degree of this polynomial}\\ \textit{is 4, and by the fundamental}\\ \textit{theorem of algebra, it can}\\ \textit{have \boxed{4} solutions at most} \end{array}$

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Doing odd jobs in America <br>2000$ How much will it take to earn?<br>I want to know time

kari74 [83]

Answer:

Below:

Step-by-step explanation:

20 Hour Per Week.....

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3 0

2 years ago

I need to know the solution to this

Alex73 [517]

There are two ways to do this but the way I prefer is to make one of the equations in terms of one variable and then 'plug this in' to the second equation. I will demonstrate

Look at equation 1, $-2x+y=10$ this can quite easily be manipulated to show $y=10+2x$ .

Then because there is a y in the second equation (and both equations are simultaneous) we can 'plug in' our new equation where y is in the second one $4x-y=-14 \Rightarrow 4x-(10+2x)=-14$ which can then be solved for x since there is only one variable $4x-10-2x=-14 \Rightarrow 2x=-4 \Rightarrow x=-2$ and then with our x solution we can work out our y solution by using the equation we manipulated $y=10+2x = 10+(2 \times -2) = 10-4=6$ .

So the solution to these equations is x=-2 when y=6

8 0

3 years ago

What is the slope-intercept form of (-2,1) and (4,6)

mash [69]

Answer:

y = $\frac{5}{6}$ x + $\frac{8}{3}$

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Calculate m using the slope formula

m = $\frac{y_{2}-y_{1} }{x_{2}-x_{1} }$

with (x₁, y₁ ) = (- 2, 1) and (x₂, y₂ ) = (4, 6)

m = $\frac{6-1}{4+2}$ = $\frac{5}{6}$ , thus

y = $\frac{5}{6}$ x + c ← is the partial equation

To find c substitute either of the 2 points into the partial equation

Using (4, 6 ), then

6 = $\frac{20}{6}$ + c ⇒ c = 6 - $\frac{20}{6}$ = $\frac{16}{6}$ = $\frac{8}{3}$

y = $\frac{5}{6}$ x + $\frac{8}{3}$ ← equation of line

4 0

3 years ago

Acone has a height of 7 ft and a radius of 4 ft. Which equation can find the volume of the cone?

Marina86 [1]

Answer:

Volume of the cone = $117.06\,feet^3$

Step-by-step explanation:

Height of the cone = $7\,feet$

Radius of the cone = $4\,feet$

Volume of a cone is:

$\pi \times r^2\times \dfrac{h}{3}$

As, $\pi =\dfrac{22}{7} =3.14$

Volume is:

$=3.14\times(4\times 4)\times \dfrac{7}{3}\\\\ =3.14\times 16 \times 2.33\\\\=3.14\times 37.28\\\\=117.06\,feet^3$

The volume of the cone is: $117.06\,feet^3$

3 0

3 years ago

Find the radius of convergence, r, of the series. ? n2xn 7 · 14 · 21 · ? · (7n) n = 1

defon

I'm guessing the series is supposed to be

$\displaystyle\sum_{n=1}^\infty\frac{n^2x^n}{7\cdot14\cdot21\cdot\cdots\cdot(7n)}$

By the ratio test, the series converges if the following limit is less than 1.

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2x^{n+1}}{7\cdot14\cdot21\cdot\cdots\cdot(7n)\cdot(7(n+1))}}{\frac{n^2x^n}{7\cdot14\cdot21\cdot\cdots\cdot(7n)}}\right|$

The first $n$ terms in the numerator's denominator cancel with the denominator's denominator:

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2x^{n+1}}{7(n+1)}}{n^2x^n}\right|$

$|x^n|$ also cancels out and the remaining factor of $|x|$ can be pulled out of the limit (as it doesn't depend on $n$ ).

$\displaystyle|x|\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2}{7(n+1)}}{n^2}\right|=|x|\lim_{n\to\infty}\frac{|n+1|}{7n^2}=0$

which means the series converges everywhere (independently of $x$ ), and so the radius of convergence is infinite.

3 0

3 years ago