First one is correct choice
The integral is path-independent if we can find a scalar function <em>f</em> such that grad(<em>f</em> ) = <em>A</em>. This requires


Take the first PDE and integrate both sides with respect to <em>x</em> to get

where <em>g</em> is assumed to be a function of <em>y</em> alone. Differentiating this with respect to <em>x</em> gives

which would mean <em>g</em> is *not* a function of only <em>y</em>, but also <em>x</em>, contradicting our assumption. So the integral is path-dependent.
Parameterize the unit circle (call it <em>C</em>) by the vector function,

with <em>t</em> between 0 and 2π.
Note that this parameterization takes <em>C</em> to have counter-clockwise orientation; if we compute the line integral of <em>A</em> over <em>C</em>, we can multiply the result by -1 to get the value of the integral in the opposite, clockwise direction.
Then

and the (counter-clockwise) integral over <em>C</em> is



and so the integral in the direction we want is -2π.
By the way, that the integral doesn't have a value of 0 is more evidence of the fact that the integral is path-dependent.
<h3>
Answer: 410 mL </h3>
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Explanation:
The ratio of A to B is 5:3
Multiply both parts by x to get 5x:3x
We need 5x parts of A and 3x parts of B
Adding those parts should get us 656 mL total
A+B = total
5x+3x = 656
8x = 656
x = 656/8
x = 82
Therefore, we need 5x = 5*82 = 410 mL of compound A
Also, we need 3x = 3*82 = 246 mL of compound B
In total, we have A+B = 410+246 = 656 mL which helps confirm the answer.
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Side note: The ratio 410:246 reduces to 5:3 after dividing both parts by 82.
The real amount is 3.333333333333333333333... so you can round it to 0.3
<1=55
<2=105
<3=55
Step-by-step explanation:
2.these are angles in a straight line so subtract 75 from 180 to get your answer
3.the angle missing from the triangle with 50 is equal to 75 since opposite angles at an x are equal
1.is equal to 55 since these are alternate or z angles with 3