15.

VARVARA [1.3K]

$6y-1.5x=8\qquad\text{add 1.5x to both sides}\\\\6y=1.5x+8\qquad\text{divide both sides by 6}\\\\y=\dfrac{1.5}{6}x+\dfrac{8}{6}\\\\y=\dfrac{15}{60}x+\dfrac{4}{3}\\\\\boxed{y=\dfrac{1}{4}x+\dfrac{4}{3}}$

8 0

3 years ago

Consider the series ∑n=1[infinity]2nn!nn. Evaluate the the following limit. If it is infinite, type "infinity" or "inf". If it d

Vikki [24]

I guess the series is

$\displaystyle\sum_{n=1}^\infty\frac{2^nn!}{n^n}$

We have

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

Recall that

$e=\displaystyle\lim_{n\to\infty}\left(1+\frac1n\right)^n$

In our limit, we have

$\dfrac n{n+1}=\dfrac{n+1-1}{n+1}=1-\dfrac1{n+1}$

$\left(\dfrac n{n+1}\right)^n=\dfrac{\left(1-\frac1{n+1}\right)^{n+1}}{1-\frac1{n+1}}$

$\implies\displaystyle2\lim_{n\to\infty}\left(\frac n{n+1}\right)^n=2\frac{\lim\limits_{n\to\infty}\left(1-\frac1{n+1}\right)^{n+1}}{\lim\limits_{n\to\infty}\left(1-\frac1{n+1}\right)}=\frac{2e}1=2e$

which is greater than 1, which means the series is divergent by the ratio test.

On the chance that you meant to write

$\displaystyle\sum_{n=1}^\infty\frac{2^n}{n!n^n}$

we have

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{2^{n+1}}{(n+1)!(n+1)^{n+1}}}{\frac{2^n}{n!n^n}}\right|=2\lim_{n\to\infty}\frac1{(n+1)^2}\left(\frac n{n+1}\right)^2$

$=\displaystyle2\left(\lim_{n\to\infty}\frac1{(n+1)^2}\right)\left(\lim_{n\to\infty}\left(\frac n{n+1}\right)^n\right)=2\cdot0\cdot e=0$

which is less than 1, so this series is absolutely convergent.

6 0

3 years ago

The half-life of Radium-224 is

Sholpan [36]

Answer:

10.98 days is 10.98/3.66 = 3 half life periods and half of the mass is lost at each half life. You can do this in your head. I have!!

Step-by-step explanation:

8 0

2 years ago

Calculate $x^{6} + (y^{2} \div z^3)\cdot w^3$ assuming that $x=2$, $y=3^2$, $z=3$, and $w=4$.

horsena [70]

Answer:

this copy and paste is to hard to read making it impossible to solve sorry.

Step-by-step explanation:

3 0

3 years ago

Y=x^2-12x+45 vertex form and coordinate vertex

zysi [14]

Best Answer

 x2-12x-45=0 Two solutions were found : x = 15 x = -3

Step by step solution :Step 1 :Skip Ad
Trying to factor by splitting the middle term

1.1 Factoring x2-12x-45

The first term is, x2 its coefficient is 1 .
The middle term is, -12x its coefficient is -12 .
The last term, "the constant", is -45 

Step-1 : Multiply the coefficient of the first term by the constant 1 • -45 = -45

Step-2 : Find two factors of -45 whose sum equals the coefficient of the middle term, which is -12 .

-45 + 1 = -44 -15 + 3 = -12 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -15 and 3
 x2 - 15x + 3x - 45

Step-4 : Add up the first 2 terms, pulling out like factors :
 x • (x-15)
 Add up the last 2 terms, pulling out common factors :
 3 • (x-15)
Step-5 : Add up the four terms of step 4 :
 (x+3) • (x-15)
 Which is the desired factorization

Equation at the end of step 1 : (x + 3) • (x - 15) = 0 Step 2 :Theory - Roots of a product :

2.1 A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

2.2 Solve : x+3 = 0

Subtract 3 from both sides of the equation :
 x = -3

Solving a Single Variable Equation :

2.3 Solve : x-15 = 0

Add 15 to both sides of the equation :
 x = 15

Supplement : Solving Quadratic Equation DirectlySolving x2-12x-45 = 0 directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

3.1 Find the Vertex of y = x2-12x-45

Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive (greater than zero).

Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.

Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.

For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 6.0000 

Plugging into the parabola formula 6.0000 for x we can calculate the y -coordinate :
 y = 1.0 * 6.00 * 6.00 - 12.0 * 6.00 - 45.0
or y = -81.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = x2-12x-45
Axis of Symmetry (dashed) {x}={ 6.00}
Vertex at {x,y} = { 6.00,-81.00}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-3.00, 0.00}
Root 2 at {x,y} = {15.00, 0.00}

3 0

3 years ago