A solid is bounded by the sphere centered at the origin of radius 5 and the infinite cylinder along the z-axis of radius 3.

(a) Write inequalities that describe the solid in Cartesian coordinates.

(b) Write inequalities that describe the solid in cylindrical coordinates.

(c) Why is this solid difficult to describe in spherical coordinates? Which of the variables ρ, θ, φ are difficult to describe? Explain.

Big Red Bookstore wants to ship books from its warehouses in Brooklyn and Queens to its stores, one on Long Island and one in Manhattan. Its warehouse in Brooklyn has 700 books and its warehouse in Queens has 2,300. Each store orders 1,500 books. It costs $1 to ship each book from Brooklyn to Manhattan and $2 to ship each book from Queens to Manhattan. It costs $5 to ship each book from Brooklyn to Long Island and $4 to ship each book from Queens to Long Island.If Big Red has a transportation budget of $8,900 and is willing to spend all of it, how many books should Big Red ship from each warehouse to each store in order to fill all the orders?

Customers arrive at a local grocery store at an average rate of 2 per minute.

(a) What is the chance that no customer will arrive at the store during a given two minute period?

(b) Since it is a “Double Coupon” day at the store, approximately 70% of the customers coming to the store carry coupons. What is the probability that during a given two-minute period there are exactly four (4) customers with coupons and one (1) without coupons?

(c) Divide one given hour into 30 two-minute periods. Suppose that the numbers of customers arriving at the store during those periods are independent of each other. Denote by X the number of the periods during which exactly 5 customers arrive at the store and 4 of them carry coupons. What is the probability that X is at least 2?

(d) What is the probability that exact 4 customers coming to the store during a given two-minute period carry coupons?

Convert the follow system of equations to standard form and solve using Simplex method to find optimal solution

Maximize 10.75x + 5.3y

constraints

0.25x + 0.6y < = 1000

0.50x + 0.1y <= 1000

0.25x + 0.3y <= 1000

x>=250, y>=250

Ti Fan and his wife have retired. They are touring the USA in a big recreational vehicle (RV). They are towing a Prius car, which they use for running errands when the RV is parked. However, towing the Prius decreases the gas mileage of the RV by 25%. Ti Fan wonders if they would save gas if his wife drives the Prius instead of towing it. The Prius gets good mileage, 50 miles per gallon. Ti Fan has never wanted to know the mileage of the RV, since it would be discouraging. Nonetheless, he looks at the odometer when he fills the RV's gas tank, then looks again when he fills it again. Towing the Prius, he drove 255 miles and burned 37 gallons of gas. How much would they save, in gallons, if they had travelled the 255 miles with his wife driving the Prius while Ti Fan drove the RV?

A school is conducting optimization studies of the resources it has. One of the principal concerns of the Director is that of the staff. The problem he is currently facing is with the number of guards in the "Emergencies" section. To this end, he ordered a study to be carried out that yielded the following results:

Time Minimum number of guards required
O to 4 40
4 to 8 80
8 to 12 100
12 to 16 70
16 to 20 120
20 to 24 50

Each guard, according to Federal labor law, must work eight consecutive hours per day. Formulate the problem of hiring the minimum number of guards that meet the above requirements, as a Linear programing model. 

For the following LP problem, determine the optimal solution by the graphical solution method.

Min Z= 3x1+2x2

Subject to 2x1+x2 >10

                   -3x1+2x2 < 6

                     X1+x2 > 6

                     X1,x1 > 0

Graph and shade the feasible region

Find a polynomial of the form f(x) = ax3 + bx2 + cx + d such that f(0) = −3, f(1) = 2, f(3) = 5, and f(4) = 0. (A graphing calculator is recommended.)

answer in fraction form.

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is complete.

reflect on the three things you learned in this section about Ivy Tech Community College that will help you become a successful student. Make sure to write complete sentences use specific examples to support specific ways that this information or resource will help you.

(2) Let ωn := e2πi/n for n = 2,3,....

1. (a) Show that the n’th roots of unity (i.e. the solutions to zn = 1) are

   ωnk fork=0,1,...,n−1.

2. (b) Show that these sum to zero, i.e.

   1+ω +ω2 +···+ωn−1 =0.nnn

3. (c) Let z◦ = r◦eiθ◦ be a given non-zero complex number. Show that the n’th roots of z◦ are

   c◦ωnk fork=0,1,...,n−1 where c◦ := √n r◦eiθ◦/n.

Let C be a plane curve parameterized by arc length by α(s), T(s) its unit tangent vector and N(s) be its unit normal vector. Show d dsN(s) = −κ(s)T(s).

PeripateticShippingLines,Inc.,isashipping company that owns n ships and provides service to n ports. Each of its ships has a schedule that says, for each day of the month, which of the ports it’s currently visiting, or whether it’s out at sea. (You can assume the “month” here has m days, for some m > n.) Each ship visits each port for exactly one day during the month. For safety reasons, PSL Inc. has the following strict requirement:

(†) No two ships can be in the same port on the same day.

The company wants to perform maintenance on all the ships this month, via the following scheme. They want to truncate each ship’s schedule: for each ship Si, there will be some day when it arrives in its scheduled port and simply remains there for the rest of the month (for maintenance). This means that Si will not visit the remaining ports on its schedule (if any) that month, but this is okay. So the truncation ofSi’s schedule will simply consist of its original schedule up to a certain specified day on which it is in a port P; the remainder of the truncated schedule simply has it remain in port P.

Now the company’s question to you is the following: Given the sched- ule for each ship, find a truncation of each so that condition (†) continues to hold: no two ships are ever in the same port on the same day.

Show that such a set of truncations can always be found, and give an algorithm to find them.

Example. Suppose we have two ships and two ports, and the “month” has four days. Suppose the first ship’s schedule is

port P1; at sea; port P2; at seaand the second ship’s schedule isat sea; port P1; at sea; port P2

Then the (only) way to choose truncations would be to have the first ship remain in port P2 starting on day 3, and have the second ship remain in port P1 starting on day 2.