Complete Question
The concentration of salt in a fluid at (x,y,z) is given by F(x,y,z)=x^2+y^4+2x^2z^2 mg/cm3 You are at the point (1,1,1).
a. In which direction should you move if you want the concentration to increase the fastest?
I keep getting <5,2,8> for this answer and it says it is incorrect
You start to move in the direction you found in part (a) at a speed of 4 cm/sec. How fast is the concentration changing?
Answer:
a
[tex]\vec \Delta F (1 ,1 , 1) = < 6 , 4 , 4 >[/tex]
b
[tex] M = 2 \sqrt{ 17 }[/tex]
Explanation:
From the question we are told that
The equation is [tex]F(x,y,z)=x^2+y^4+2x^2z^2[/tex]
differentiating with respect to x
[tex]F_x (x, y, z) = 2x + 4xz^2[/tex]
differentiating with respect to y
[tex]F_y (x, y, z) = 4y^3[/tex]
differentiating with respect to z
[tex]F_z (x, y, z) = 4x^2z[/tex]
Gnerally the rate of change of the salt concentration is mathematically represented as
[tex]\vec \Delta F (x ,y , z) = <F_x , F_y , F_z >[/tex]
=> [tex]\vec \Delta F (x ,y , z) = <2x + 4xz^2 ,4y^3 , 4x^2z >[/tex]
At the point (1,1,1)
[tex]\vec \Delta F (1 ,1 , 1) = <2(1) + 4(1)(1)^2 ,4(1)^3 , 4(1)^2(1) >[/tex]
=> [tex]\vec \Delta F (1 ,1 , 1) = < 6 , 4 , 4 >[/tex]
generally rate at which the concentration is changing is mathematically represented as
[tex]M= \sqrt{ 6^2 +4^2 + 4^2}[/tex]
=> [tex]M= \sqrt{ 36 + 16 + 16}[/tex]
=> [tex]M= \sqrt{68}[/tex]
=> [tex]M = \sqrt{ 4 * 17 }[/tex]
=> [tex]M= 2 \sqrt{ 17 }[/tex]
