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2 years ago

Walter’s sink can hold 12 gallons of water. If it is 85% full, how many gallons of water are in the sink?

Mathematics

1 answer:

katen-ka-za [31]2 years ago

4 0

Answer:

10.2 gallons

Step-by-step explanation:

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For each of the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential f

Phantasy [73]

The key idea is that, if a vector field is conservative, then it has curl 0. Equivalently, if the curl is not 0, then the field is not conservative. But if we find that the curl is 0, that on its own doesn't mean the field is conservative.

$\mathrm{curl}\vec F=\dfrac{\partial(5x+10y)}{\partial x}-\dfrac{\partial(-6x+5y)}{\partial y}=5-5=0$

We want to find $f$ such that $\nabla f=\vec F$ . This means

$\dfrac{\partial f}{\partial x}=-6x+5y\implies f(x,y)=-3x^2+5xy+g(y)$

$\dfrac{\partial f}{\partial y}=5x+10y=5x+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=10y\implies g(y)=5y^2+C$

$\implies\boxed{f(x,y)=-3x^2+5xy+5y^2+C}$

so $\vec F$ is conservative.

$\mathrm{curl}\vec F=\left(\dfrac{\partial(-2y)}{\partial z}-\dfrac{\partial(1)}{\partial y}\right)\vec\imath+\left(\dfrac{\partial(-3x)}{\partial z}-\dfrac{\partial(1)}{\partial z}\right)\vec\jmath+\left(\dfrac{\partial(-2y)}{\partial x}-\dfrac{\partial(-3x)}{\partial y}\right)\vec k=\vec0$

Then

$\dfrac{\partial f}{\partial x}=-3x\implies f(x,y,z)=-\dfrac32x^2+g(y,z)$

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

$\dfrac{\partial f}{\partial z}=1=\dfrac{\mathrm dh}{\mathrm dz}\implies h(z)=z+C$

$\implies\boxed{f(x,y,z)=-\dfrac32x^2-y^2+z+C}$

so $\vec F$ is conservative.

$\mathrm{curl}\vec F=\dfrac{\partial(10y-3x\cos y)}{\partial x}-\dfrac{\partial(-\sin y)}{\partial y}=-3\cos y+\cos y=-2\cos y\neq0$

so $\vec F$ is not conservative.

$\mathrm{curl}\vec F=\left(\dfrac{\partial(5y^2)}{\partial z}-\dfrac{\partial(5z^2)}{\partial y}\right)\vec\imath+\left(\dfrac{\partial(-3x^2)}{\partial z}-\dfrac{\partial(5z^2)}{\partial x}\right)\vec\jmath+\left(\dfrac{\partial(5y^2)}{\partial x}-\dfrac{\partial(-3x^2)}{\partial y}\right)\vec k=\vec0$

Then

$\dfrac{\partial f}{\partial x}=-3x^2\implies f(x,y,z)=-x^3+g(y,z)$

$\dfrac{\partial f}{\partial y}=5y^2=\dfrac{\partial g}{\partial y}\implies g(y,z)=\dfrac53y^3+h(z)$

$\dfrac{\partial f}{\partial z}=5z^2=\dfrac{\mathrm dh}{\mathrm dz}\implies h(z)=\dfrac53z^3+C$

$\implies\boxed{f(x,y,z)=-x^3+\dfrac53y^3+\dfrac53z^3+C}$

so $\vec F$ is conservative.

4 0

3 years ago

What line intersect to make a right angle

bazaltina [42]

2 lines intersect at one point and stop there to form an L shaped angle they also form a square when you combine another right angle and they measure 90 degrees. Hope this helps!

3 0

2 years ago

If y’all can help with this one it’s imagine 2 i’m trying to figure out

Sindrei [870]

Answer:

I think it would be D. I'm not sure but I think so. I thought C at first but (go to explanation)

Step-by-step explanation:

if image 1 is the original, then image 2 would be it reflected over the y axis

C says over the line y=0 which would be a horizontal line. x=0 would be a vertical line aka the y axis. I believe its D.

8 0

2 years ago

Are the two right triangles trs and wuv congruent? If so name the congruence postulate that applies

Nataly_w [17]

<span>Two figures are said to be congruent when they have the same shape and size or if one object is a mirror image of the other object.<span> The right triangles trs and wuv are congruent when they have equal lengths of the sides and having equal sides, angles should as well be equal with the two triangles.</span></span>

8 0

2 years ago

Roger makes 70 gallons of pink paint by mixing 21 gallons of white paint. what part of every gallon is from red paint?

forsale [732]

There are 49 gallons of red paint.
I figured it out by doing the following:

$70 \: - \: 21 \: = \: 49$
If you subtract the gallons of pink paint (21) from the total number of gallons (70), then you get the number of gallons of red paint (49) because 49 + 21 = 70. I hope that helps :)

8 0

2 years ago