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stira [4]
3 years ago
12

Three concert tickets cost $45 what is the cost per ticket

Mathematics
1 answer:
11Alexandr11 [23.1K]3 years ago
4 0

Answer: 15

Step-by-step explanation:

45$ = 3 Tickets

45 / 3 = 1 Ticket

1 Ticket = 15

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(X^2+y^2+x)dx+xydy=0<br> Solve for general solution
aksik [14]

Check if the equation is exact, which happens for ODEs of the form

M(x,y)\,\mathrm dx+N(x,y)\,\mathrm dy=0

if \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.

We have

M(x,y)=x^2+y^2+x\implies\dfrac{\partial M}{\partial y}=2y

N(x,y)=xy\implies\dfrac{\partial N}{\partial x}=y

so the ODE is not quite exact, but we can find an integrating factor \mu(x,y) so that

\mu(x,y)M(x,y)\,\mathrm dx+\mu(x,y)N(x,y)\,\mathrm dy=0

<em>is</em> exact, which would require

\dfrac{\partial(\mu M)}{\partial y}=\dfrac{\partial(\mu N)}{\partial x}\implies \dfrac{\partial\mu}{\partial y}M+\mu\dfrac{\partial M}{\partial y}=\dfrac{\partial\mu}{\partial x}N+\mu\dfrac{\partial N}{\partial x}

\implies\mu\left(\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}\right)=M\dfrac{\partial\mu}{\partial y}-N\dfrac{\partial\mu}{\partial x}

Notice that

\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}=y-2y=-y

is independent of <em>x</em>, and dividing this by N(x,y)=xy gives an expression independent of <em>y</em>. If we assume \mu=\mu(x) is a function of <em>x</em> alone, then \frac{\partial\mu}{\partial y}=0, and the partial differential equation above gives

-\mu y=-xy\dfrac{\mathrm d\mu}{\mathrm dx}

which is separable and we can solve for \mu easily.

-\mu=-x\dfrac{\mathrm d\mu}{\mathrm dx}

\dfrac{\mathrm d\mu}\mu=\dfrac{\mathrm dx}x

\ln|\mu|=\ln|x|

\implies \mu=x

So, multiply the original ODE by <em>x</em> on both sides:

(x^3+xy^2+x^2)\,\mathrm dx+x^2y\,\mathrm dy=0

Now

\dfrac{\partial(x^3+xy^2+x^2)}{\partial y}=2xy

\dfrac{\partial(x^2y)}{\partial x}=2xy

so the modified ODE is exact.

Now we look for a solution of the form F(x,y)=C, with differential

\mathrm dF=\dfrac{\partial F}{\partial x}\,\mathrm dx+\dfrac{\partial F}{\partial y}\,\mathrm dy=0

The solution <em>F</em> satisfies

\dfrac{\partial F}{\partial x}=x^3+xy^2+x^2

\dfrac{\partial F}{\partial y}=x^2y

Integrating both sides of the first equation with respect to <em>x</em> gives

F(x,y)=\dfrac{x^4}4+\dfrac{x^2y^2}2+\dfrac{x^3}3+f(y)

Differentiating both sides with respect to <em>y</em> gives

\dfrac{\partial F}{\partial y}=x^2y+\dfrac{\mathrm df}{\mathrm dy}=x^2y

\implies\dfrac{\mathrm df}{\mathrm dy}=0\implies f(y)=C

So the solution to the ODE is

F(x,y)=C\iff \dfrac{x^4}4+\dfrac{x^2y^2}2+\dfrac{x^3}3+C=C

\implies\boxed{\dfrac{x^4}4+\dfrac{x^2y^2}2+\dfrac{x^3}3=C}

5 0
3 years ago
A car is traveling along a test track in the desert at a constant rate of 85 feet per second The test track has a length of 10,0
jeyben [28]

Answer:

foot marker 320 at t=0

Step-by-step explanation:

8 sec x 85 ft/s =680 ft

1,000ft - 680 ft = 320 ft

4 0
2 years ago
Which expression is equivalent to 7/32 + 2/32
Marta_Voda [28]

Answer:

36 square root 2 also known as C

Step-by-step explanation:

8 0
2 years ago
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I need help solving this inverse function. Does anybody know how to do it? I'm almost gonna be having my finals so I wanna be re
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3 0
3 years ago
I have 120 cm of framing material to make a picture frame, which will be most pleasing to the eye if its height is 2/3 of its wi
Ilia_Sergeevich [38]

 

The formula for perimeter is:

<span>P = 2 h +  2 w</span>

where P is perimeter = 120 cm, h is height, w is width

we are also given that: h = (2/3) w, therefore:

P = 2 (2/3) w + 2 w = 120

(4/3) w + 2 w = 120

w = 36 cm

 

Therefore l is:

l = (2/3) w = 24 cm

 

<span>Hence the dimension should be 24cm by 36 cm</span>

6 0
3 years ago
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