Robert's Responses in WyzAnt Answers

Differentiate both sides of xe^y+ye^z+2lnx-2-3ln2 = 0 with respect to y,

xe^y + e^z + ye^z ∂z/∂y = 0

Plug in (1, ln 2, ln 3),

2 + 3 + 3ln2 ∂z/∂y = 0

∂z/∂y = -5/(3ln2) <==Answer

∂x/∂t = 1/[2sqrt(1+t)] = 1/4 at t = 3

∂y/∂t = 3

dT/dt = (∂T/∂x)(∂x/∂t) + (∂T/∂y)(∂y/∂t) = (4)(1/4) + 2(3) = 7 degrees of C/sec <==Answer

Method I. Using linear programming concept

Since the maximum value must be reached at the boundary of the circle,  draw a tangent line with the circle such that the tangent line has a slope of -4/3. Therefore, the point of tangency can be written as (4a, 3a), where a > 0. Plug into the equation of the circle: (3a)^2 + (4a)^2 = 1 => a = 1/5

fmax = f(4/5, 3/5) = 3(3/5) + 4(4/5) = 5 <==Answer

Method II. Using Lagrange multiplier

g(x, y) = 3y+4x + λ(x^2+y^2-1)

g'_x = 4 + 2λx = 0 => x = -2/λ, (x > 0)

g'_y = 3 + 2λy = 0 => y = -3/(2λ), (y > 0)

x^2+y^2 = (-2/λ)^2 + (-3/(2λ))^2 = 1 => λ = -2.5

x = 4/5, y = 3/5

fmax = f(4/5, 3/5) = 3(3/5) + 4(4/5) = 5 <==Answer

a)

Horizontal direction: v_o cos45 t = 10 ......(1)

Vertical direction: v_o sin45 t - (1/2)g t² = 0 ......(2)

Since sin45 = cos45,

10 = (1/2)gt² ==> t = √(20/g)

Plug in (1),

v_o√(10/g) = 10

Solve for v_o,

v_o = √(10g)

b)

Horizontal direction: v_o cosθ t = 6 ......(1)

Vertical direction: v_o sinθ t - (1/2)g t² = 0 =>Since t ≠ 0,  t = 2v_o sinθ/g......(2)

Plug in (2) in (1),

v_o²sin2θ/g = 6 => 10sin2θ = 6, since v_o² = 10g.

sin2θ = 3/5

θ = (1/2)arcsin(3/5)

or

2θ = pi - arcsin(3/5)

θ = pi/2 - (1/2)arcsin(3/5)

Yes, arcsin(x) = sin^-1(x)

Concept: Three non-colinear points determine a plane.

Ideas of approach: Find the equation of the line first, and pick two points from the line plus the point P.

The equation of the line has a direction of <1, 1, -1> X <2, -1 ,3> = <2, -5, -3>

Pick one point from the line: <1, 1, 0>

So, the equation of the line is r = <1, 1, 0> + t<2, -5, -3>

Let t = 1 to get another point on the line <3, -4, -3>

With the three non-colinear points: <1, 1, 0>, <3, -4, -3> and <-1, 2, 1>, you can get the equation of the plane: x - 2y + 4z = -1.