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DIA [1.3K]
3 years ago
12

Along with the flowers, Samantha bought 4 packages of glass beads and 2 vases. The vases cost $3.59 each and the total sales tax

was $1.34. The total amount she paid was $28.50, including sales tax. Explain a strategy she could use to find the cost of 1 package of glass beads.<br />
Mathematics
1 answer:
zmey [24]3 years ago
4 0

Samantha can find the total of the vases then add that and the sales tax. Once she has added those numbers up, she should subtract that from the total cost of the items. Once Samantha has subtracted, she should divide the answer by 4 to get the cost of an individual bag of beads

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Liono4ka [1.6K]

Answer:

a) f (x,y,z)= xy^2\sin(z)

b) \int_C F \cdot dr =0

Step-by-step explanation:

Recall that given a function f(x,y,z) then \nabla f = (\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}). To find f, we will assume it exists and then we will find its form by integration.

First assume that F = \nabla f. This implies that

\frac{\partial f}{\partial x} = y^2\sin(z) if we integrate with respect to x we get that

f(x,y,z) = xy^2\sin(z) + g(y,z) for some function g(y,z). If we take the derivative of this equation with respect to y, we get

\frac{\partial f}{\partial y} = 2xy\sin(z) + \frac{\partial g}{\partial y}

This must be equal to the second component of F. Then

2xy\sin(z) + \frac{\partial g}{\partial y}=2xy\sin(z)

This implies that \frac{\partial g}{\partial y}=0, which means that g depends on z only. So f(x,y,z) = xy^2\sin(z) + g(z)

Taking the derivative with respect to z and making it equal to the third component of F, we get

xy^2\cos(z)+\frac{dg}{dz} = xy^2\cos(z)

which implies that \frac{dg}{dz}=0 which means that g(z) = K, where K is a constant. So

f (x,y,z)= xy^2\sin(z)

b) To evaluate \int_C F \cdot dr we can evaluate it by using f. We can calculate the value of f at the initial and final point of C and the subtract them as follows.

\int_C F \cdot dr = f(r(\pi))-f(r(0))

Recall that r(\pi) = (\pi^2, 0, \pi) so f(r(\pi)) = \pi^2\cdot 0 \cdot \sin(\pi) = 0

Also r(0) = (0, 0, 0) so f(r(0)) = 0^2\cdot 0 \cdot \sin(0) = 0

So \int_C F \cdot dr =0

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