Production Planning A bakery sells Boston cream pies and carrot cakes. Each Boston cream pie requires 30 minutes preparation time, 30 minutes baking time, and 15 minutes for finishing. Each carrot cake requires 45 minutes preparation time, 50 minutes baking time, and 10 minutes for finishing. (a) Write a matrix T representing the required time for preparation, baking, and finishing for the Boston cream pies and the carrot cakes. (b) The bakery receives an order for 20 Boston cream pies and 8 carrot cakes for a large party. Find a matrix S so that either ST or TS gives the total preparation, baking, and finishing times required to fill this order. (c) What is the total baking time? What is the total finishing time?
Production Planning A bakery sells Boston cream pies and carrot cakes. Each Boston cream pie requires 30 minutes preparation time, 30 minutes baking time, and 15 minutes for finishing. Each carrot cake requires 45 minutes preparation time, 50 minutes baking time, and 10 minutes for finishing. (a) Write a matrix T representing the required time for preparation, baking, and finishing for the Boston cream pies and the carrot cakes. (b) The bakery receives an order for 20 Boston cream pies and 8 carrot cakes for a large party. Find a matrix S so that either ST or TS gives the total preparation, baking, and finishing times required to fill this order. (c) What is the total baking time? What is the total finishing time?
Production Planning A bakery sells Boston cream pies and carrot cakes. Each Boston cream pie requires 30 minutes preparation time, 30 minutes baking time, and 15 minutes for finishing. Each carrot cake requires 45 minutes preparation time, 50 minutes baking time, and 10 minutes for finishing.
(a) Write a matrix T representing the required time for preparation, baking, and finishing for the Boston cream pies and the carrot cakes.
(b) The bakery receives an order for 20 Boston cream pies and 8 carrot cakes for a large party. Find a matrix S so that either ST or TS gives the total preparation, baking, and finishing times required to fill this order.
(c) What is the total baking time? What is the total finishing time?
Q2: Using the Laplace transform, find the solution for the following equation
y"" +y" = 6et + 6t + 6. Suppose zero initial conditions (y"" (0) = y"(0) = y'(0) = y(0) = 0).
1- Let A = {A1, A2, ...), in which A, A, = 0, when i j.
a) Is A a π-system? If not, which element(s) should be added to A to become a π-system?
b) Prove that σ(A) consists of the finite or countable unions of elements of A; i.c., A E σ(A) if and
only if there exists finite or countable sequence {n} such that A = U₁An (Hint: Let F be such
class; prove that F is a σ-filed containing A.)
c) Let p ≥ 0 be a sequence of non-negative real numbers with Σip₁ = 1. Using p₁'s, how do you
construct a probability measure on σ(A)? (Hint: use extension theorem.)
2- Construct an example for which P(lim sup A,) = 1 and P(lim inf An) = 0.
3. Let
f(z) =
sin (22) + cos (T2)
2(22+1)(z+1)
Compute f(z)dz over each of the contours/closed curves C1, C2, C3 and C4 shown
below.
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