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.

Write the equation of the tangent line to the graph of ?(?) = (2)/(3−?) at the point where x = 4. You must use the limit definition for any derivatives and show each process by step. Use proper notation.

Find the domain for each function please explain. f(x)=10x^2 + x f(x)= -2/x^2

Discuss a real time application of first order differential equation [Answer should include literature review and real applications maximum 200 words].

a) Given a vector field à = zỹ +(3y + 2)2 î in cartesian coordinates, determine whether it is solenoidal (V · À = 0), conservative (D x X = 0)

I Div x A (Cylinderical Coordinates)

ii) Calculate integral A*dl , where the contour C is the unit circle (r=1) traversed in anticlockwise direction

Find the value of k such that the graph of y = f(x) has no vertical asymptote given by x = 2 where

f(x) = (4x 3 − 4x 2 + kx + 14)/ 4x 2 − 12x + 8 .

Then find all the intercepts, asymptotes, local extreme values, points of inflection, monotonicity intervals, concavity intervals. Finally sketch the graph of y = f(x).

Find the general solution for differential equation

x^3y'''-(3x^2)y''+6xy'-6y=0, y(1)=2, y'(1)=1, y''(1)=-4  

1) Determine the angle between vectors:

U = <2, -3, 4> and V= <-1, 3, -2>

2) determine the distance between line and point

P: -2x+3y-4z =2

L: 3x – 5y+z =1

3) Determine the distance between the line L and the point A given by

L; (x-1)/2 = (y+2)/5 = (z-3)/4 and A (1, -1,1)

4) Find an equation of the line given by the points A, B and C.

A (2, -1,0), B (-2,4,-1) and C ( 3,-4,1)

5) Determine whether the lines are parallel, perpendicular or neither.

(x-1)/2 = (y+2)/5 = (z-3)/4 and (x-2)/4 = (y-1)/3 = (z-2)/6

6) A) Find the line intersection of vector planes given by the equations

-2x+3y-z+4=0 and 3x-2y+z=-2

B) Given U = <2, -3, 4> and V= <-1, 3, -2>

Find

a. U . V

b. U x V

7) Find the angle between the planes:

3x -5y+7z -4=0 and 3x -2y+5z +3 =0

Find the area under the graph of f over the interval

left bracket 7 comma 9 right bracket[7,9].

​f(x)equals=StartSet Start 2 By 2 Matrix 1st Row 1st Column 4 x plus 9 comma 2nd Column for x less than or equals 8 2nd Row 1st Column 81 minus 5 x comma 2nd Column for x greater than 8 EndMatrix

The area is

nothing.

2)

Find the area of the region bounded by the graphs of the given equations.

yequals=x plus 6x+6​,

yequals=x squaredx2

The area is

nothing.

​(Type an integer or a simplified​ fraction.)