The radius of a right circular cone is increasing at a rate

Question

1. Use the the chain rule to find ∂z/∂s and ∂z/∂t where 

Z=X² + xy + y², x = 3s + 4t, y= 5s + 6t

2. Let W(S, t) = F(u(s,t), v(s,t)) where

u(1,0) = -5, u_s(1, 0) = 3,u_t(1, 0) = -4
v(1,0) = 2, v_s(1, 0) = -4, v_t(1, 0) = 6
F_u(-5, 2) = -3, F_u(-5, 2) = -2

W_s(1,0) = ?

W_t(1,0) = ?

3. Suppose z = x² sin y, x = 2s² + 4t², y = -6st.

A. Use the chain rule to find ∂z/∂s and ∂z/∂t as functions of x, y, s and t.

B. Find the numerical values of ∂z/∂s and ∂z/∂t when (S, t) = (-1, -3).

4.

Let

w = 3xy - yz + 4xz, x = st, y _{= e}st_{,  z} = t²

Compute

and 

∂w/∂s (-2, 5) =

∂w/∂t (-2 ,5) =

5. The radius of a right circular cone is increasing at a rate of 3 inches per second and its height is decreasing at a rate of 2 inches per second. At what rate is the volume of the cone changing when the radius is 40 inches and the height is 20 inches?

cubic inches per second

6. In a simple electric circuit, Ohm's law states that V = I R. where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. Assume that, as the battery wears out. the voltage decreases at 0.03 volts per second and, as the resistor heats up, the resistance is increasing at 0.01 ohms per second. When the resistance is 400 ohms and the current is 0.01 amperes. at what rate is the current changing?

7. Find dy/dx in terms of x and y if x⁴y - x - 5y - 5 = 0.

8. Find dy/dx in terms of x and y if x ln y + y⁴ = 6 In X.

9. Find the slope of the tangent to the curve x² + xy + y² = 7 at (1, 2)

slope =