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Harlamova29_29 [7]
3 years ago
12

joann is having a party. she wants to make punch. the recipe for punch uses 4 pints of orange juice, 5 cups of apple juice, 1/2

galloon of lemonade and 2 quarts of pinapple juice. joann says her recipe will make 30 cups. is she correct?
Mathematics
1 answer:
Aleks [24]3 years ago
7 0
4 pints= 8 cups
+
1/2 gallon= 8 cups
+
2 quarts= 8 cups
+
5 cups
=29
no, she is off by a cup
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\vec F(x,y,z)=(x^2+y)\,\vec\imath+(y^2+x)\,\vec\jmath+ze^z\,\vec k

We want to find f(x,y,z) such that \nabla f=\vec F. This means

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f(x,y,z)=e^z(z-1)+g(x,y)

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Then

f(x,y,z)=e^z(z-1)+\dfrac{x^3}3+xy+h(y)

and differentiating both sides with respect to y gives

y^2+x=x+\dfrac{\mathrm dh}{\mathrm dy}\implies\dfrac{\mathrm dh}{\mathrm dy}=y^2\implies h(y)=\dfrac{y^3}3+C

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\boxed{f(x,y,z)=e^z(z-1)+\dfrac{x^3}3+xy+\dfrac{y^3}3+C}

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and \vec F does the same amount of work over both of the other paths.

In part (b), I don't know what is meant by "df/dt for F"...

In part (c), you're asked to find the work over the 2 parts (call them L_1 and L_2) of the given path. Using the fundamental theorem makes this trivial:

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