Solve the following ODE using Laplace Transforms
?̈+?̇+3?=0;?(0)=1; ?̇(0)=2

Solve the heat equation (in one dimensional case) for c^2=9, the following boundary and initial conditions:

u(0,t)=u(2Pie,t)=0

u(x,0)=5sinx = 2sin5x

GTA Construction Corporation constructed two buildings near the San Andreas fault line. The probability that either of these buildings will experience an earthquake is 4.6 percent. However, if one building experiences an earthquake, the probability that the second building will experience an earthquake is 57 percent. What is the probability (in percent) that both buildings will experience earthquake damage?

IMB Computing creates motherboards for cellphones at their campuses in Seattle and San Diego. The company is worried about computer hackers and hired a consultant to evaluate their risk. The consultant estimated that the San Diego campus has a 12.1 percent chance of being hacked. The consultant also noted that the Seattle location has a 24.4 percent chance of digital hacking. IMB would asks the consultant, what is the probability (in percent) that both campuses will suffer hacking related crime in any given year?

Hishiba Company assembles hard drives and has plants in both the South and the North, spaced about 3,000 miles apart and connected by light rail. Hishiba is worried about local rain causing flooding at their plants. The probability that in any given year a flood will damage the North plant 5.1 percent. The probability that in any given year a flood will damage the South plant is 13 percent. What is the probability (in percent) that at least one of the plants will be damaged by flood in any given year?

Your job is to fully staff your facility at the lowest cost. The facility must have at least 2 people w working from 6am-8pm Monday thru Friday and at least 1 person working from 10am to 6 pm on Saturdays. Nobody works on Sundays. Full time staff must work 8 hours a day, five days a week. Part time staff work 4 hours a day, 5 days a week. Nobody is allowed to work overtime. All employees receive the same hourly rate. What is the number of part time and full time employees that fully staffs the operation with the smallest total labor cost?

"Find the retail cost of gasoline (per unit volume) in your area and find the cost of electricity for residential users in your area. (a) Compare the cost of energy (per MJ) from gasoline and from electricity (b) The efficiency of a gasoline automobile is about 20% and the efficiency of a battery electric vehicle is about 85%. Discuss the relative transportation costs for gasoline and electric vehicles."

a real sequence x_n is defined inductively by x₁ =1 and x_n+1 = sqrt(x_n +6) for every n belongs to N

a) prove by induction that x_n is increasing and x_n <3 for every n belongs to N

b) deduce that x_n converges and find its limit

1(a) If ut − kuxx = f, vt − kvxx = g, f ≤ g, and u ≤ v at x = 0, x = l and t = 0, prove that u ≤ v for 0 ≤ x ≤ l, 0 ≤ t < ∞.

(b) If vt − vxx ≥ cos x for −π/2 ≤ x ≤ π/2, 0 < t < ∞, and if v(−π/2, t) ≥ 0, v(π/2, t) ≥ 0 and v(x, 0) ≥ cos x, use part (a) to show that v(x, t) ≥ (1 − e −t ) cos x.

If you needed to compare two simulated scenarios, and in one of them a self-driving car is braking up to -3 meters per second squared, in another up to -8 meters per second squared (unit of acceleration), which one do you think is more risky? Please explain your answer or how you arrived to this conclusion.

Consider an algebra where the vector space is ℝ³ and the multiplication of vectors is the conventional cross product you learned as a beginning physics student. Find the structure constants of this algebra.

Find Taylor series expansion for sin^2(z)

Solve by reduction of order

xy" - xy' + y = 0

1. An experiment was conducted to test the effect of different lighting systems and the presence or absence of a watchman on the average number of car burglaries per month at a parking garage. The data are in the following table and also in the file GARAGE, where the variables are named BURGLAR, LIGHTING and WATCHMAN.

1. Using the Lenth procedure, find which effect(s) (if any) appear significant.

effects

2. Interpret all of these significant effects.
3. Permanently upgrading the lighting in the garage will cost $700 per month. Hiring awatchman will cost $3000 per month. It is estimated that lawsuits and loss of business cost on average $2000 per burglary. Is it worth it to hire a watchman, install lights or both?