The answer is 33 pie mm^3
The differential equation
![M(x,y) \, dx + N(x,y) \, dy = 0](https://tex.z-dn.net/?f=M%28x%2Cy%29%20%5C%2C%20dx%20%2B%20N%28x%2Cy%29%20%5C%2C%20dy%20%3D%200)
is considered exact if
(where subscripts denote partial derivatives). If it is exact, then its general solution is an implicit function
such that
and
.
We have
![M = \tan(x) - \sin(x) \sin(y) \implies M_y = -\sin(x) \cos(y)](https://tex.z-dn.net/?f=M%20%3D%20%5Ctan%28x%29%20-%20%5Csin%28x%29%20%5Csin%28y%29%20%5Cimplies%20M_y%20%3D%20-%5Csin%28x%29%20%5Ccos%28y%29)
![N = \cos(x) \cos(y) \implies N_x = -\sin(x) \cos(y)](https://tex.z-dn.net/?f=N%20%3D%20%5Ccos%28x%29%20%5Ccos%28y%29%20%5Cimplies%20N_x%20%3D%20-%5Csin%28x%29%20%5Ccos%28y%29)
and
, so the equation is indeed exact.
Now, the solution
satisfies
![f_x = \tan(x) - \sin(x) \sin(y)](https://tex.z-dn.net/?f=f_x%20%3D%20%5Ctan%28x%29%20-%20%5Csin%28x%29%20%5Csin%28y%29)
Integrating with respect to
, we get
![\displaystyle \int f_x \, dx = \int (\tan(x) - \sin(x) \sin(y)) \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20f_x%20%5C%2C%20dx%20%3D%20%5Cint%20%28%5Ctan%28x%29%20-%20%5Csin%28x%29%20%5Csin%28y%29%29%20%5C%2C%20dx)
![\implies f(x,y) = -\ln|\cos(x)| + \cos(x) \sin(y) + g(y)](https://tex.z-dn.net/?f=%5Cimplies%20f%28x%2Cy%29%20%3D%20-%5Cln%7C%5Ccos%28x%29%7C%20%2B%20%5Ccos%28x%29%20%5Csin%28y%29%20%2B%20g%28y%29)
and differentiating with respect to
, we get
![f_y = \cos(x) \cos(y) = \cos(x) \cos(y) + \dfrac{dg}{dy}](https://tex.z-dn.net/?f=f_y%20%3D%20%5Ccos%28x%29%20%5Ccos%28y%29%20%3D%20%5Ccos%28x%29%20%5Ccos%28y%29%20%2B%20%5Cdfrac%7Bdg%7D%7Bdy%7D)
![\implies \dfrac{dg}{dy} = 0 \implies g(y) = C](https://tex.z-dn.net/?f=%5Cimplies%20%5Cdfrac%7Bdg%7D%7Bdy%7D%20%3D%200%20%5Cimplies%20g%28y%29%20%3D%20C)
Then the general solution to the exact equation is
![f(x,y) = \boxed{-\ln|\cos(x)| + \cos(x) \sin(y) = C}](https://tex.z-dn.net/?f=f%28x%2Cy%29%20%3D%20%5Cboxed%7B-%5Cln%7C%5Ccos%28x%29%7C%20%2B%20%5Ccos%28x%29%20%5Csin%28y%29%20%3D%20C%7D)
Answer:
i. Colonel is about 201 feet away from the fire.
ii. Sarge is about 125 feet away from the fire.
Step-by-step explanation:
Let the Colonel's location be represented by A, the Sarge's by B and that of campfire by C.
The total angle at the campfire from both the Colonel and Sarge =
+ ![34^{0}](https://tex.z-dn.net/?f=34%5E%7B0%7D)
= ![93^{0}](https://tex.z-dn.net/?f=93%5E%7B0%7D)
Thus,
<CAB =
-
= ![31^{0}](https://tex.z-dn.net/?f=31%5E%7B0%7D)
<CBA =
-
= ![56^{0}](https://tex.z-dn.net/?f=56%5E%7B0%7D)
Sine rule states;
=
= ![\frac{c}{Sin C}](https://tex.z-dn.net/?f=%5Cfrac%7Bc%7D%7BSin%20C%7D)
i. Colonel's distance from the campfire (b), can be determined by applying the sine rule;
= ![\frac{c}{Sin C}](https://tex.z-dn.net/?f=%5Cfrac%7Bc%7D%7BSin%20C%7D)
= ![\frac{242}{Sin 93^{0} }](https://tex.z-dn.net/?f=%5Cfrac%7B242%7D%7BSin%2093%5E%7B0%7D%20%7D)
= ![\frac{242}{0.9986}](https://tex.z-dn.net/?f=%5Cfrac%7B242%7D%7B0.9986%7D)
cross multiply,
b = ![\frac{0.8290*242}{0.9986}](https://tex.z-dn.net/?f=%5Cfrac%7B0.8290%2A242%7D%7B0.9986%7D)
= 200.8993
Colonel is about 201 feet away from the fire.
ii. Sarge's distance from the campfire (a), can be determined by applying the sine rule;
= ![\frac{c}{Sin C}](https://tex.z-dn.net/?f=%5Cfrac%7Bc%7D%7BSin%20C%7D)
= ![\frac{242}{Sin 93^{0} }](https://tex.z-dn.net/?f=%5Cfrac%7B242%7D%7BSin%2093%5E%7B0%7D%20%7D)
= ![\frac{242}{0.9986}](https://tex.z-dn.net/?f=%5Cfrac%7B242%7D%7B0.9986%7D)
cross multiply,
a = ![\frac{0.5150*242}{0.9986}](https://tex.z-dn.net/?f=%5Cfrac%7B0.5150%2A242%7D%7B0.9986%7D)
= 124.8073
Sarge is about 125 feet away from the fire.
Answer:WTVY
Step-by-step explanation:
Answer:
1.A+B=2y²+3x-x²+3x²-y²=y²+3x+2x²
2.A-B=2y²+3x-x²-3x²+y²=3y²+3x-4x²
3.B+C=3x²-y²+5x²-3xy=8x²-3xy-y²
4.B-C=3x²-y²-5x²+3xy=3xy-2x²-y²
5.A+B+C=2y²+3x-x²+3x²-y²+5x²-3xy
=y²+3x+2x²+5x²-3xy=y²+3x-3xy+7x²
6.A+B-C=y²+3x+2x²-5x²+3xy=y²+3x+3xy-3x²