5. (9 points) Show that the function u(x, y) = x³ − 3xy² + y is harmonic and determine it's harmonic conjugate.

Find the mass of the part of the paraboloid z = 1-(x² + y²), 0 < x < 1,0 < y < 1if the surface density p = xy.

Question 3 Solve simultaneous equations to find values of x and y that work for both the triangles below simultaneously.

) Solve the given initial value problem using the method of Laplace transforms. y^{\prime \prime}+4 y=4 t^{2}-4 t+10 ; \quad y(0)=0, y^{\prime}(0)=3

For each of the situations below, determine which notation(s), fraction, decimal, or percent, is most appropriate.

A warehouse, whose dimensions are shown below, needs stucco. The front and back of the warehouse each have a roll up door measuring 14 ft by 8 ft. The side of the warehouse facing the parking lot has a window measuring 45 in by 32 in and an entry door measuring 4 ft by 7 ft. The other side of the warehouse has no window or door. Use the given information to answer the questions. Each tab shows a different view of the warehouse. a) Assuming the roof, windows, and doors require no stucco, what is the area in square feet that needs stucco? (Do not round any intermediate computations and give your answer as a whole number.) ) The stucco to be used is sold in bags. Each bag contains enough stucco to cover 160 ft“. Assume there is no stucco yet and partial bags cannot be bought. How many bags will need to be bought in order to apply stucco to the warehouse? ) What is the total cost of the stucco needed for the warehouse if each bag costs $14.50?

BE is parallel to CD. AB = 9 cm, BC =3 cm, CD-7 cm, AE=6 cm. (a) Calculate the length of ED. (b) Calculate the length of BE.

The SIR model for modeling infectious diseases is given by \left\{\begin{array}{l} S^{\prime}(t)=-\frac{\beta}{N} I(t) S(t) \\ I^{\prime}(t)=\frac{\beta}{N} I(t) S(t)-\gamma I(t) \\ R^{\prime}(t)=\gamma I(t) \end{array}\right. where N is the total number of people in a given population, S(t) is the number of susceptible people attime t, I(t) is the number of infected people at time t, R(t) is the number of removed people at time t,and 3,y quantify rates of spread and removal. 1. Given the parameters 3 = 3, y = .2, use Euler's method with Ax = .1 to solve the SIR model withinitial conditions S(0) = 9997, I(0) = 3, R(0) = 0 for 0 <t < 20. 2. Given the parameters 3 = 1, y = .2, us Euler's method with Ax = .1 to solve the SIR model withinitial conditions S(0) = 9997, I(0) = 3, R(0) = 0 for 0 < t< 20. 3. Given the parameters 3 = 1,7 = .8, us Euler's method with Ax = .1 to solve the SIR model withinitial conditions S(0) = 9997, I(0) = 3, R(0) = 0 for 0 < t < 20. Your final excel file should have plots for all three scenarios above. Discuss the differences between thefound for the different values of the parameters and how this is related to infectious diseasesolutionsyoumodeling.

Suppose BD bisects ABC and E on the ray opposite to BD. Prove tha ABD = CBD.

For the following DE, state where on the ty-plane the hypotheses of the Existence and Uniqueness Theorem for first order s is satisfied. Sketch a picture of this 2-dimensional region. y^{\prime}=\frac{\ln (2 t y)}{4-t^{2}+y^{2}}