Show that 0.252525 is a rational number

olya-2409 [2.1K]

A)

$\bf x^3+y^3=6xy\\\\
-----------------------------\\\\
3x^2+3y^2\frac{dy}{dx}=6\left( 1\cdot y+x\frac{dy}{dx} \right)\\\\\\ 3\left( x^2+ y^2\frac{dy}{dx}\right)=6\left( 1\cdot y+x\frac{dy}{dx} \right)
\\\\\\
 x^2+ y^2\frac{dy}{dx}=2y+2x\frac{dy}{dx}\implies x^2+ y^2\frac{dy}{dx}-2y-2x\frac{dy}{dx}=0
\\\\\\
\cfrac{dy}{dx}(y^2-2x)=2y-x^2\implies \cfrac{dy}{dx}=\cfrac{2y-x^2}{y^2-2x}$

B)

$\bf \left. \cfrac{dy}{dx}=\cfrac{2y-x^2}{y^2-2x} \right|_{3,3}\implies 
\begin{cases}
x=3\\
y=3
\end{cases}\implies -1
\\\\
-----------------------------\\\\
y-{{ y_1}}={{ m}}(x-{{ x_1}})\implies y-3=-1(x-3)\\
\qquad \uparrow\\
\textit{point-slope form}$

solve for "y", and that'd be the equation of the tangent at 3,3

C) recall, the tangent is horizontal when the slope is 0, thus

$\bf \cfrac{dy}{dx}=\cfrac{2y-x^2}{y^2-2x}\implies 0=\cfrac{2y-x^2}{y^2-2x}\implies 0=2y-x^2\implies \cfrac{x^2}{2}=\boxed{y}
\\\\\\
now \qquad x^3+y^3=6xy\implies x^3+\left( \boxed{\cfrac{x^2}{2}} \right)^3=6x\left( \boxed{\cfrac{x^2}{2}} \right)
\\\\\\
x^3+\cfrac{x^6}{8}=3x^3\implies \cfrac{8x^3+x^6}{8}=3x^3
\\\\\\
8x^3+x^6-24x^3=0\implies x^6-16x^3=0
\\\\\\
x^3(x^2-16)=0\implies x=\{0,\pm 4\}$

so.. x =0 is not exactly in the first quadrant, x = -4 isn't either

so. the only choice is really x = 4... what's "y" when x = 4? well, just plug that into the $\bf x^3+y^3=6xy\qquad or \qquad \cfrac{x^2}{2}=y$

and solve for "y"

4 0

3 years ago

The faces of a tetrahedron die are numbered 1 to 4. if each of the 4 sides is equally likely to land down, what is the probabili

Hoochie [10]

The probability distribution for the face landing down = 4/15

<h3>What is the probability?</h3>

The area of mathematics known as probability deals with numerical descriptions of how likely it is for an event to happen or for a claim to be true. A number between 0 and 1 is the probability of an event, where, broadly speaking, 0 denotes the event's impossibility and 1 denotes its certainty.

<h3>According to the given information:</h3>

If we assign 1 probability x

then 2 has probability 1/2 x.

Similarly,

then 3 has probability = 1/2((1/2)*x) = (1/4)*x

so

then 4 has probability = 1/2((1/4)*x) = (1/8)*x

These probabilities need to sum to 1.

Hence:

$x+\frac{1}{2} x+\frac{1}{4} x+\frac{1}{8} x=1$

or equivalently

(15/8)*x = 1

x = 8/15

the probability distribution for the face landing down = (1/2)*(8/15)

= 8/30

= 4/15

To know more about probability visit:

brainly.com/question/1581057

#SPJ4

I understand that the question you are looking for is:

A tetrahedral die has four faces, numbered 1-4. If the die is weighted in such a way that each number is twice as likely to land facing down as the next number (1 twice as likely as 2, 2 twice as likely as 3, and so on) what is the probability distribution for the face landing down?

7 0

2 years ago

Jake and each of his brothers choose a fraction between 0 and 1.Jake chooses 3/4,Aaron chooses 9/10, and Simon choses 4/12. Whic

Bad White [126]

Answer:

c. 9/12, 90/100,1/3

Step-by-step explanation:

3/4, *3=9/12

9/10 *10 =90/100

4/12 divided into 4 = 1/3

8 0

3 years ago

Read 2 more answers

A tile store charges $607.50 to install 135 square feet of tile. Assuming they charge the same rate

jasenka [17]

Answer:

the tile is 4.50 per square foot to install

it will cost 3780.00 to install 840 square feet

Step-by-step explanation:

divide 607.50 by 135 to find out the price per square foot to install

multiply the price per square foot by the number of square feet you are installing so 4.50 X 840

7 0

4 years ago

What is m<BCD<br>OA) 80°<br>OB) 90°<br>C) 100°<br>D) 110°

wlad13 [49]

Answer:

D

Step-by-step explanation:

The external angle of a triangle is equal to the sum of the 2 opposite interior angles.

∠ BCD is an exterior angle of the triangle, thus

∠ BCD = 50° + 60 = 110° → D

6 0

3 years ago

Show that 0.252525 is a rational number​

Show that 0.252525 is a rational number