Missing information:
How fast is the temperature experienced by the particle changing in degrees Celsius per meter at the point
![P = (\frac{1}{2}, \frac{\sqrt 3}{2})](https://tex.z-dn.net/?f=P%20%3D%20%28%5Cfrac%7B1%7D%7B2%7D%2C%20%5Cfrac%7B%5Csqrt%203%7D%7B2%7D%29)
Answer:
![Rate = 0.935042^\circ /cm](https://tex.z-dn.net/?f=Rate%20%3D%200.935042%5E%5Ccirc%20%2Fcm)
Step-by-step explanation:
Given
![P = (\frac{1}{2}, \frac{\sqrt 3}{2})](https://tex.z-dn.net/?f=P%20%3D%20%28%5Cfrac%7B1%7D%7B2%7D%2C%20%5Cfrac%7B%5Csqrt%203%7D%7B2%7D%29)
![T(x,y) =x\sin2y](https://tex.z-dn.net/?f=T%28x%2Cy%29%20%3Dx%5Csin2y)
![r = 1m](https://tex.z-dn.net/?f=r%20%3D%201m)
![v = 2m/s](https://tex.z-dn.net/?f=v%20%3D%202m%2Fs)
Express the given point P as a unit tangent vector:
![u = \frac{\sqrt 3}{2}i - \frac{1}{2}j](https://tex.z-dn.net/?f=u%20%3D%20%5Cfrac%7B%5Csqrt%203%7D%7B2%7Di%20-%20%5Cfrac%7B1%7D%7B2%7Dj)
Next, find the gradient of P and T using: ![\triangle T = \nabla T * u](https://tex.z-dn.net/?f=%5Ctriangle%20T%20%3D%20%5Cnabla%20T%20%2A%20u)
Where
![\nabla T|_{(\frac{1}{2}, \frac{\sqrt 3}{2})} = (sin \sqrt 3)i + (cos \sqrt 3)j](https://tex.z-dn.net/?f=%5Cnabla%20T%7C_%7B%28%5Cfrac%7B1%7D%7B2%7D%2C%20%5Cfrac%7B%5Csqrt%203%7D%7B2%7D%29%7D%20%20%3D%20%28sin%20%5Csqrt%203%29i%20%2B%20%28cos%20%5Csqrt%203%29j)
So: the gradient becomes:
![\triangle T = \nabla T * u](https://tex.z-dn.net/?f=%5Ctriangle%20T%20%3D%20%5Cnabla%20T%20%2A%20u)
![\triangle T = [(sin \sqrt 3)i + (cos \sqrt 3)j] * [\frac{\sqrt 3}{2}i - \frac{1}{2}j]](https://tex.z-dn.net/?f=%5Ctriangle%20T%20%3D%20%5B%28sin%20%5Csqrt%203%29i%20%2B%20%28cos%20%5Csqrt%203%29j%5D%20%2A%20%20%5B%5Cfrac%7B%5Csqrt%203%7D%7B2%7Di%20-%20%5Cfrac%7B1%7D%7B2%7Dj%5D)
By vector multiplication, we have:
![\triangle T = (sin \sqrt 3)* \frac{\sqrt 3}{2} - (cos \sqrt 3) \frac{1}{2}](https://tex.z-dn.net/?f=%5Ctriangle%20T%20%3D%20%28sin%20%5Csqrt%203%29%2A%20%20%5Cfrac%7B%5Csqrt%203%7D%7B2%7D%20-%20%28cos%20%5Csqrt%203%29%20%20%5Cfrac%7B1%7D%7B2%7D)
![\triangle T = 0.9870 * 0.8660 - (-0.1606 * 0.5)](https://tex.z-dn.net/?f=%5Ctriangle%20T%20%3D%200.9870%20%2A%200.8660%20-%20%28-0.1606%20%2A%200.5%29)
![\triangle T = 0.9870 * 0.8660 +0.1606 * 0.5](https://tex.z-dn.net/?f=%5Ctriangle%20T%20%3D%200.9870%20%2A%200.8660%20%2B0.1606%20%2A%200.5)
![\triangle T = 0.935042](https://tex.z-dn.net/?f=%5Ctriangle%20T%20%3D%200.935042)
Hence, the rate is:
<h2>
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♥Answer♥</h3>
Santiago Chile (AP)
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The answer is 244 because you add 35 to 287 then subtract 78 and it equals 244
The answer is (C) the middle one