Demand for walnut fudge ice cream at the Sweet Cream Dairy can be approximated by a normal distribution with a mean of 17 gallons per week and a standard deviation of 3.2 gallons per week. The new manager desires a service level of 90 percent. Lead time is two days, and the dairy is open seven days a week. (Hint: Work in terms of weeks.)

a-1. If an ROP model is used, what ROP would be consistent with the desired service level? (Do not round intermediate calculations. Round your final answer to 2 decimal places.)

ROP ______ gallons

a-2. How many days of supply are on hand at the ROP, assuming average demand? (Do not round intermediate calculations. Round your final answer to 2 decimal places.)

Days _______

b-1. If a fixed-interval model is used instead of an ROP model, what order size would be needed for the 90 percent service level with an order interval of 7 days and a supply of 8 gallons on hand at the order time? (Do not round intermediate calculations. Round your final answer to the nearest whole number.)

Order size _______ gallons

b-2. What is the probability of experiencing a stockout before this order arrives? (Do not round intermediate calculations. Round your final answer to the nearest whole percent. Omit the "%" sign in your response.)

Probability _________ %

c. Suppose the manager is using the ROP model described in part a. One day after placing an order with the supplier, the manager receives a call from the supplier that the order will be delayed because of problems at the supplier’s plant. The supplier promises to have the order there in two days. After hanging up, the manager checks the supply of walnut fudge ice cream and finds that 2 gallons have been sold since the order was placed. Assuming the supplier’s promise is valid, what is the probability that the dairy will run out of this flavor before the shipment arrives? (Do not round intermediate calculations. Round your final answer to the nearest whole percent. Omit the "%" sign in your response.)

Risk probability _________ %

U(C1, C2, C3, C4, C5) = C1∙C2∙C3∙C4∙C5

As a mathematical function, does U have a maximum or minimum value? What values of Ci correspond to the minimum value of U? What values of Ci correspond to the maximum value of U? Do these values of Ci make sense from an economic standpoint?

Now let us connect the idea of economic utility to actual dollar values. To keep the values more manageable, we will use household income rather than the entire state budget, and retail costs and measures rather than industrial ones. Find the Median Household Income for Mesa, AZ for the most recent year possible. Then find the dollar cost in Mesa, AZ for a Penny, a pound of Ground Beef, a pair of Jeans, fresh Orange Juice, and a Movie Ticket. (Entertainment is often used as a stand-in for Climate.) A Cost-of- Living Index is a good place to find much of this data. Record these prices as P1, P2, P3, P4, and P5 respectively.

Construct an equation using Median Income, the Ci and Pi values that illustrates how much of each resource the Median Household can afford to purchase. Given this restriction, do the maximum or minimum values of U change? Do the values of Ci that give the maximum or minimum values change? What are these new values? How should the Median Household budget its Income so as to maximize its Economic Utility?

Write up your findings in a paper that you could turn in to an employer. Be sure to show all your work. Include any appropriate references as well as any computational devices used.

* Solve the questions, make the table, and the graph.

Students will be asked to formulate, define, and interpret mathematical modeling (particularly ordinary differential equation) which involves real engineering applications and related to their majoring (E.g. Newton’s law cooling/warming, mixture problem, radioactive decay, spring-mass system, series circuit, deflection of the beam, etc.).. The selected model should be solved analytically using any methods that have been learnt in the mathematic lecture. just give me an example with related topic and how to solve it using math

Show that the quotient ring Q[x]/(x2 − 3) is isomorphic to a subfield of the real numbers R.

Consider the given matrix.

Find the eigenvalues. (Enter your answers as a comma-separated list.)

λ = 3i,−3i

(I got these right)
Find the eigenvectors of the matrix. (Enter your answers in order of the corresponding eigenvalues, from smallest to largest by real part, then by imaginary part.)

I can't seem to get the eigenvectors right.

Given the differential equation dy/dx =x. With intitial condition y(1)=0.5. Use eulers method with dx = 0.1 , to approximate the value of y when x=1.8

Find the general solution of the given system.

[x(t), y(t)]= _____________, _______________

(6c1​+8c2​)10​sin(6t)+(6c2​+8c1​)10​cos(6t), c1​cos(6t)+c2​sin(6t)

^above is the answer I got, which is incorrect.

State the dual of the Theorem below.

Let a non-degenerate plane conic touch the sides BC, CA, and AB of a triangle ABC in R² at the points P, Q, and R respectively. Then AP, BQ, and CR are concurrent.

Using Runge-Kutta method of order 4 to approximate y(1) with step size h = 0.1 and h = 0.2 respectively (keep 8 decimals):

dy/dx = x + arctan y, y(0) = 0.

Solutions: when h = 0.1, y(1) = 0.70398191. when h = 0.2, y(1) = 0.70394257.

7. OR. If you prefer, calculate Proj_CS(A)b using the approach of Example 8 on page 266 instead. You should get an infinite number of solutions for x to the normal equation. Any solution for x will work, and then calculate Ax to get Proj_CS(A)
   • For either approach, briefly explain why Proj_CS(A)b is the closest vector in your plane to the vector (See Theorem 5.15 if needed.)
8. Choose 5 arbitrary unique vectors in R² that are not on the same line or scalar multiples of each other. What are your 5 vectors in R²?
   • Use normal equations (see page 265, if needed) to determine the equation of the least square regression line for this set of 5 points.
9. Choose any 2 of the 5 vectors in the above problem that will form a nonstandard basis for R², which we will call B’. (12 points for entire problem)
   • What are your two vectors in B’?
   • If B is the standard basis for R² (I.e. B ={(1.0). (0.1)}, determine the transition matrix (called P^-1 in our text) from B to B’.
   • Use P^-1 to calculate the coordinates for (1,5) with respect to the basis B’.
10. Investigate the following linear differential equation: y’’ + 4y = 0; Solutions {sin(2x), cos(2x)}

• Verify that each solution satisfies the differential equation.
• Use the Wronskian to verify that the solution set is linearly independent.
• Write the general solution of the differential equation.

True or False? Why?

Σ n = 1, ∞ fn(x) converges uniformly on A <=> for all n in N (natural numbers), there exists Mn > 0 such that |fn(x)| <= Mn for all x in A and Σ n = 1, ∞ Mn converges.