Please show all work step by step so i can understand

Find and classify each critical point (as relative maximum, relative minimum, or saddle point) of f(x,y)=2x^3+3x^2+y^2-36x+8y+1

A water balloon launcher (a is used to propel a water balloon from a sidewalk that is 48 ft above a level parking lot on campus. The water balloon leaves the launcher at a speed of 64 ft/sec at an angle of 30 degrees with the horizontal. a) When will the water balloon hit the pavement? b) How far will the balloon be horizontally from the launch point at that time?

Suppose that Line 1 contains the point P1 = (1,2,3) and the vector V1 = <2,1,-2> is parallel to Line 1, and also that Line 2 contains the point P2 = (4,0,9) and that the vector V2 = <-2,-1,2> is parallel to line 2. Find the distance between Line 1 and Line 2.

The Jones family is buying a new house at the price of $165,000. They will finance it with a twenty-year mortgage that has an interest rate of 8%.

(a)Assuming that the family can make a $39,000 down payment, what will their monthly mortgage payment be?

(b)If the family could increase the down payment by $10,000,then how much would their monthly mortgage payment be?

(c)In total, how much money can the family save by making the larger down payment

An investor has a certain number of units of a new currency whose price is fluctuating periodically. The number of units that the investor holds, N(x), as a function of time x in days is given by: N(x) = 2x π + 5 The price of each unit, P(x), as a function of time x in days is given by: P(x) = sin x 2 + 1 The total value of the investment, T(x), as a function of time x in days is given by the number of units × the price of each unit: T(x) = N(x) · P(x) a) Find the total value of the investment at x = 5π 2 days. b) Find the derivative of T(x). c) What is the instantaneous rate of change of the total value of the investment at x = 5π 2 days?

You are making a box by removing the corners from an 16cm by 10cm rectangular box. How much do you need to remove to make a box of maximum value?

Please show explanation

Set-up surface area of y = cos 2x rotated about x-axis [0, π/4] and sketch the surface

2. Given the System of Equations:

3x+2y+z+20w= 6

x+2y+z+10w=0

x+y+z+6w=2

2x+2y+z+15w=3

a) Use your calculator to solve, leaving solution in parametric form

b) Find the specific solution when y = 6

c) Perform, BY HAND, a full check of this particular solution

y(x)= C₁e^2x +C₂e^-x +C₃Cos(x)+C₄Sin(x)-4x⁵+10x⁴+20x³+30x²-450x+255

?(0) = ? ′(0) = ? ′′(0) = ?''' (0) = 0

Find the solution to the initial value problem by plugging in the initial conditions to the general solution.

a. Find ? ′(?), ? ′′(?), and ? (3) (?). Make sure to calculate for both pieces of the general solution.

b. Plug in initial condition and find system of coefficients.

c. Solve the system of coefficients. (If you find this problem in the text, the answer in the back is incorrect.)

d. Write out general solution ?(?) with filled in ?1, ?2, ?3, ?4 values.

The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 468 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 9 dollar increase in rent. Similarly, one additional unit will be occupied for each 9 dollar decrease in rent. What rent should the manager charge to maximize revenue?

An open cone is filled completely with water and is oriented with its vertex facing downward. The cone has a base diameter of 6 inches and a height of 12 inches. Assume the cone starts leaking water from its vertex at a constant rate of 3? in3/hr.

a.Find an equation for the volume of water in this cone in terms of the height only.

b.Find the height of the water in the cone four hours after the water started leaking.

c.Using correct units, find the rate of change in the height of the water in the cone at the time from 2b.

d.Using correct units, find the change of the cross-sectional area of the top of the water in the cone at the time from 2b.

1. Find a closed form expression for the MacLaurin series for f(x) = sinh(3x)

2. Find a closed form expression for the Taylor series for f(x) = 4e^2x expanded at a=3

A rectangular swimming pool is 5 ft deep, 10 ft wide, and 15 ft long. The pool is filled with water to 1 ft below the top.  If the weight density of water is 62.4 lb / ft³ and if x = 0 corresponds to the bottom of the tank, then which of the following represents the work done (in ft-lb) in pumping all the water into a drain at the top edge of the pool?

Consider the cost of assigning a task to an individual as shown in the table below. It is assumed that each individual can be assigned to at most one task, and each task can be assigned to at most one individual. The objective is to minimize the cost of assignments.

(a) Write down the linear programming formulation of this problem. (i.e., write down the objective function and constraints – do not use a tableau.)

(b) Using the Hungarian Algorithm, solve this assignment problem (i.e., the problem described on the previous page). Please show the order in which the tableaus are used!

(c) State the optimal values of the variables and the optimal objective function.

Let J be the antipodal of A in the circumcircle of triangle ABC. Let M be the midpoint of side BC. Let H be the orthocenter of triangle ABC. Prove that H, M, and J are collinear.