A company manufactures three types of luggage: economy, standard, and deluxe. The
company produces 1000 pieces each month. The cost for making each type of luggage is
$20 for the economy model, $30 for the standard model, and $40 for the deluxe model.
The manufacturer has a budget of $25,700. Each economy piece requires 6 hours of
labor, the standard model requires 8 hours of labor and the deluxe requires 12 hours of
labor. The manufacturer has a total of 7400 hours of labor available each month. If the
manufacturer sells all of the luggage produced, exhausts his entire budget and uses all
available hours of labor, how many pieces of each can be produced?

a) Write the system of equations for this information. Identify variables.

b) Write the augmented matrix for this information.

c) Solve using the Gauss-Jordan elimination method. Label each answer.

Use your knowledge of second-order systems forced by a sinusoidal function to solve (a) y '''' + y ′′ = sin x.

Hint: Try to integrate twice immediately. Extra hint: Your solution should involve four constants.

(b) Instead consider y'''' + y ′′ = x sin x. Note that ∫ x s sin s s = sin x − x cos x + c, where c is a constant.

Use Newton's forward and backward difference formula to construct interpolating polynomials of degree 1,2, and 3 from the following data and approximate the function at the specified value using these polynomials:

f(0.43) if

f(0) =1

f(0.25) = 1.6487,

f(0.5) = 2.7182,

and f(0.75) = 4.4819

Find y as a function of x if

y′′′−11y′′+30y′=0

y(0)=7,  y′(0)=9,  y′′(0)=4.

Find the solution of y′′+15y′+54y=144e−3t

with y(0)=3 and y′(0)=2.

Question 1 [35 marks] A foundry that specializes in producing custom blended alloys has received an order for 1 000 kg of an alloy containing at least 5% chromium and not more than 50% iron. Four types of scrap which can be easily acquired can be blended to produce the order. The cost and metal characteristics of the four scrap types are given below: Scrap type Item 1 2 3 4 Chromium 5% 4% - 8% Iron 40% 80% 60% 32% Cost per kg R6 R5 R4 R7 The purchasing manager has formulated the following LP model: Minimise COST = 6M1 + 5M2 + 4M3 + 7M4 subject to 0,05M1 + 0,04M2 + 0,08M4 ≥ 50 (CHRM) 0,40M1 + 0,80M2 + 0,60M3 + 0,32M4 ≤ 500 (IRON) M1 + M2 + M3 + M4 = 1000 (MASS) and all variables ≥ 0, where Mi = number of kg of scrap type i purchased, i=1,2,3,4. (a) Solve this model using LINDO or SOLVER. (b) Write down the foundry's optimal purchasing plan and cost. 4 PBA4804 OCTOBER/NOVEMBER 2019 PORTFOLIO EXAMINATION [TURN OVER] Based on your LINDO or SOLVER solution answer the following questions by using only the initial printout of the optimal solution. (This means that you may not change the relevant parameters in the model and do reruns.) (c) How good a deal would the purchasing manager need to get on scrap type 1 before he would be willing to buy it for this order? (d) Upon further investigation, the purchasing manager finds that scrap type 2 is now being sold at R5,40 per kg. Will the purchasing plan change? By how much will the cost of purchasing the metals increase? (e) The customer is willing to raise the ceiling on the iron content in order to negotiate a reduction in the price he pays for the order. How should the purchasing manager react to this? (f) The customer now specifies that the alloy must contain at least 6% chromium. Can the purchasing manager comply with this new specification? Will the price charged for the order change?

Show that g(x) = 1/2sin(x-1) has a unique fixed point on the interval [-1,1]

Estimate the number of iterations required to achieve 10^-2 accuracy

Find all homomorphism

(1) f : z -> z_5

(2) f: z_5 -> z_5

(3) f: z_3 -> S_3

1) what are Cauchy’s Integral Theorem and Cauchy's integral formula

2) Explain about the consequences and applications of these theorems.

3) Explain about different types of singularities and the difference between Taylor series and Laurent series.

Given the plot of y=f(x) below, find the plot of y=f−1(x).

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. A curve starts at the point left-parenthesis negative 1 comma 0 right-parenthesis, rises at an increasing rate from left to right and passes through left-parenthesis 1 comma 1 right-parenthesis and left-parenthesis 4 comma 6 right-parenthesis.

Select the correct answer below:

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. A curve starts at the point left-parenthesis 0 comma negative 1 right-parenthesis, rises at a decreasing rate from left to right and passes through left-parenthesis 1 comma 1 right-parenthesis and left-parenthesis 4 comma 3 right-parenthesis.

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. A curve starts at the point left-parenthesis 0 comma 1 right-parenthesis, rises at a decreasing rate from left to right and passes through left-parenthesis 1 comma 3 right-parenthesis and left-parenthesis 4 comma 5 right-parenthesis.

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. A curve starts at the point left-parenthesis 1 comma 0 right-parenthesis, falls at an increasing rate from right to left and passes through left-parenthesis negative 1 comma negative 1 right-parenthesis and left-parenthesis negative 3 comma negative 4 right-parenthesis.

A coordinate plane has a horizontal x-axis labeled from negative 7 to 7 in increments of 1 and a vertical y-axis labeled from negative 7 to 7 in increments of 1. From left to right, a curve falls shallowly in quadrant 2, passing through the points left-parenthesis negative 4 comma 6 right-parenthesis and left-parenthesis negative 2 comma 2 right-parenthesis, crosses the positive y axis at 0.5, and ends at left-parenthesis 1 comma 0 right-parenthesis.

Can someone explain Bessel's Equations in Layman's terms

If a function f(x) is odd about a point, say (a,0), on the x-axis what exactly does this mean? How would you relate f(x values to left of a) to f(x values to right of a)?

Similarly, if a function f(x) is even about a point, say (a,0), on the x-axis what exactly does this mean? How would you relate f(x values to left of a) to f(x values to right of a)?

I understand what is meant by odd and even functions about the origin, I just want to make sure I understand how a function can be odd or even about a point on the x axis. Can you also have odd and even functions about points on the y-axis? How would you express these?

Find a Formula for the degree 2 Taylor polynomial T₂(x,y) at (a,b)=(pi/2,0). Do not simplify your formula. Use a 3d graphing tool to verify T2(x,y) does a good job of approximating f(x,y) near (a,b)

3. What are the necessary and sufficient conditions for a bipartite graph to have a perfect matching? Justify your answer.

4. Illustrate Lemma 3.1.21 using the Peterson graph.