Solve the following initial value problem using Laplace transforms:
y000 + y0 = et, y(0) = y0(0) = y00(0) = 0.

Solve the initial value problem once using power series method and once using the characteristic method. Please show step for both 3) 3y”−y=0, y(0)=0,y’(0)=1

Note that 3y” refers to it being second order differential and y’ first

Market Research:

You have an apple orchard. You let people pick their own apples. When you let people pick a small bag for $6, there were 6,316 small bags sold. You do some market research and find that a $0.25 increase in price means 235 fewer small bags of apples sold. Recall that revenue is quantity sold X Price per quantity. If we let x be the number of $0.25 increases or decreases from $6 and R be revenue, then answer the following questions:

A) What equation represents what you expect the revenue to be based on your research?

B) Based on your answer in A, if you charge $5.75 (x= -1) how much do you expect the revenue to be?

C) At what price or prices will you have $30,000 in sales, If your expectations listed above are true?

I would appreciate any and all help. If you give the answers, could you please show work and explain how you got them? I did my best in trying to re-write the questions (my professor doesnt have great grammar.)

Let (R 3 , ×) be the set of 3d vectors equipped with the operation of vector crossproduct. Which of the following properties does this operation satisfy (give proofs in all cases)?

(a) has identity element(s)? (If so, determine all identity elements.)

(b) has idempotent element(s)? (If so, determine all idempotent elements.)

(c) commutative?

(d) associative?

2. Find the point on the line 6x+y=9 that is closest to the point (-3,1).

a. Find the objective function.

b. Find the constraint.

c. Find the minimum (You need to specify your method)

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.