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3 years ago

Do these three sides make a triangle? Is it unique? 1.4 miles, 1.8 miles, and 4.5 miles

Mathematics

1 answer:

elena-14-01-66 [18.8K]3 years ago

8 0

Answer:

No It Is Not A Triangle

Step-by-step explanation:

To be a triangle the 2 shorter sides have to add up greater than the longest side, 1.4+1.8=3.2 which is not grater than 4.5

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(4x+3y^2)dx+2xy*dy=0 by using integrating fector

Nataly_w [17]

$\underbrace{(4x+3y^2)}_M\,\mathrm dx+\underbrace{2xy}_N\,\mathrm dy=0$

The ODE is exact if $\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}$ .

$M_y=6y$
$N_x=2y$

This is not the case, so look for an integrating factor $\mu(x)$ such that

$\dfrac\partial{\partial y}\mu M=\dfrac\partial{\partial x}\mu N$

Since $\mu$ is a function of $x$ only, you have

$\mu M_y=\mu'N+\mu N_x\implies\dfrac{\mu'}\mu=\dfrac{M_y-N_x}N\implies\mu=\exp\left(\displaystyle\int\frac{M_y-N_x}N\,\mathrm dx\right)$

So, the integrating factor is

$\mu=\exp\left(\displaystyle\int\frac{6y-2y}{2xy}\,\mathrm dx\right)=\exp\left(2\int\frac{\mathrm dx}x\right)=x^2$

Now the ODE can be modified as

$\underbrace{(4x^3+3x^2y^2)}_{M^*}\,\mathrm dx+\underbrace{2x^3y}_{N^*}\,\mathrm dy=0$

Check for exactness:

${M^*}_y=6x^2y$
${N^*}_x=6x^2y$

so the modified ODE is indeed exact.

Now, you're looking for a solution of the form $\Psi(x,y)=C$ , since differentiating via the chain rule yields

$\dfrac{\mathrm d}{\mathrm dx}\Psi(x,y)=\Psi_x+\Psi_y\dfrac{\mathrm dy}{\mathrm dx}=0$

Matching up components, you would have

$\Psi_x=M^*=4x^3+3x^2y^2$
$\displaystyle\int\Psi_x\,\mathrm dx=\int(4x^3+3x^2y^2)\,\mathrm dx$
$\Psi=x^4+x^3y^2+f(y)$

Differentiate this with respect to $y$ to get

$\Psi_y=2x^3y+f'(y)=2x^3y=N^*$
$f'(y)=0\implies f(y)=C_1$

So the solution here is

$\Psi(x,y)=x^4+x^3y^2+C_1=C\implies x^4+x^3y^2=C$

Just for a final check, take the derivative to get back the original ODE:

$\dfrac{\mathrm d}{\mathrm dx}[x^4+x^3y^2]=\dfrac{\mathrm d}{\mathrm dx}C$
$4x^3+3x^2y^2+2x^3y\dfrac{\mathrm dy}{\mathrm dx}=0$
$4x+3y^2+2xy\dfrac{\mathrm dy}{\mathrm dx}=0$
$(4x+3y^2)\,\mathrm dx+2xy\,\mathrm dy=0$

so the solution is correct.

7 0

4 years ago

Combine like terms to simplify:

dimaraw [331]

Answer: D. -y-x

Step-by-step explanation: combine like terms

4 0

4 years ago

Read 2 more answers

What is an equation that has an infinite number of solutions

Maurinko [17]

2x + 7 = x + x + 7 is an example of an equation with an infinite number of solutions, meaning any number could be the answer.

5 0

3 years ago

Seven times the difference of a number and 9is equal to 2

goldfiish [28.3K]

The number . . . . . N
The difference of the number and 9 . . . N-9
Seven times that difference . . . . . . . 7(N-9)

You said that 7(N-9) = 2 .

Divide each side by 7 : N - 9 = 2/7

Add 9 to each side: N = 9 and 2/7

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4 years ago

3^(2x+3)=6^(x+2) Please help

Nitella [24]

3^(2x+3)=6^(x+2)=STUDY

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4 years ago