Final answer:
The positive value of (dy)/(dt) is 9 when x=8 and y=6, found by differentiating the equation z²=x²+y² with respect to t and substituting the given values.
Step-by-step explanation:
To find the positive value of (dy)/(dt) given the information that z²=x²+y², that (dx)/(dt)=2, and that (dz)/(dt)=7 when x=8 and y=6, we can differentiate the equation z²=x²+y² with respect to t to obtain:
2z(dz/dt) = 2x(dx/dt) + 2y(dy/dt).
Substitute the given values into the differentiated equation:
2z(7) = 2(8)(2) + 2(6)(dy/dt).
With the values of x and y, we can find z by calculating the square root of x² + y²:
z = √(x²+y²) = √(8²+6²) = √(64+36)= √100=10.
Now, substituting z=10 into the equation:
2(10)(7) = 2(8)(2) + 2(6)(dy/dt)
140 = 32 + 12(dy/dt)
140 = 32 + 12(dy/dt)
(dy/dt)=(140 - 32)/12
(dy/dt)=108/12
(dy/dt)=9.
Therefore, the positive value of (dy)/(dt) is 9 when x=8 and y=6.