Mathematics: Ms. Annand is getting drinks ready for a party. She has one gallon of fruit punch, two pints of apple juice, and a quart of grape soda. How much is she bringing all together for drinks to the party, measured in cups?

Ms. Annand is getting drinks ready for a party.

Ответ:

22kbachep9igbr

16.03.2022

Mathematics

4+4+16=24

Step-by-step explanation:

16 cups in a gallon, 4 cups in two pints, and 4 cups in a quart. Hope this helps!

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Ответ:

genyjoannerubiera

16.06.2021

Mathematics

$\displaystyle\int_{\mathcal C}2xyz\,\mathrm dx+x^2z\,\mathrm dy+x^2y\,\mathrm dz=\int_{\mathcal C}\underbrace{(2xyz,x^2z,x^2y)}_{\mathbf F}\cdot\underbrace{(\mathrm dx,\mathrm dy,\mathrm dz)}_{\mathrm d\mathbf r}$

The fact that $\mathcal C$ is not described exactly is a clue to try to show the vector field $\mathbf F$ is conservative. That means you need to show there is some scalar function

for which $\mathbf F=\nabla f$ ; that is, the following set of PDEs is satisfied:

$\dfrac{\partial f}{\partial x}=2xyz$
$\dfrac{\partial f}{\partial y}=x^2z$
$\dfrac{\partial f}{\partial z}=x^2y$

Integrating both sides of the first PDE with respect to

gives

Differentiating with respect to

gives

$\dfrac{\partial f}{\partial y}=x^2z+\dfrac{\partial g}{\partial z}=x^2z\implies\dfrac{\partial g}{\partial z}=0\implies g(y,z)=h(z)$

Differentiating with respect to

gives

$\dfrac{\partial f}{\partial z}=x^2y+\dfrac{\mathrm dh}{\mathrm dz}=x^2y\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C$

So the potential scalar function is

f(x,y,z)=x^2yz+C

By the fundamental theorem of calculus, we have for any oriented simple curve $\mathcal C$ parameterized by a vector-valued function $\mathbf r$ , starting at $\mathbf a$ and terminating at $\mathbf b$ ,

$\displaystyle\int_{\mathcal C}\nabla f\cdot\mathrm d\mathbf r=f(\mathbf r(\mathbf b))-f(\mathbf r(\mathbf a))$

In this case, $\mathbf a=(1,2,4)$ and $\mathbf b=(1,4,16)$ , so the line integral has a value of

$\displaystyle\int_{\mathcal C}2xyz\,\mathrm dx+x^2z\,\mathrm dy+x^2y\,\mathrm dz=f(1,4,16)-f(1,2,4)=64-8=56$

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Ms. Annand is getting drinks ready for a party. She has one gallon of fruit punch, two pints of apple juice, and a quart of grape soda. How much is she bringing all together for drinks to the party, measured in cups?

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