Question

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 1,000 books and its warehouse in Queens has 2,000. Each store orders 1,500 books. It costs $5 to ship each book from Brooklyn to Long Island and $1 to ship each book from Brooklyn to Manhattan. It costs $4 to ship each book from Queens to Long Island and $2 to ship each book from Queens to Manhattan. (a) If Red has a transportation budget of $8,540 and is willing to spend all of it, how many books should Red ship from each warehouse to each store in order to fill all the orders? (Assume Red spends the entire transportation budget.)1. Brooklyn to Long Island books2. Brooklyn to Manhattan books3. Queens to Long Island books4. Queens to Manhattan books

Answer 1

The number of books Red Bookstore shipped from:

(1) Brooklyn to Long Island = 270

(2) Brooklyn to Manhattan = 730

(3) Queens to Long Island = 1230

(4) Queens to Manhattan = 770

Explanation:

Define the variables as follows:

B₁ = Number of books shipping from Brooklyn to Long Island

B₂ = Number of books shipping from Brooklyn to Manhattan

Q₁ = Number of books shipping from Queens to Long Island

Q₂ = Number of books shipping from Queens to Manhattan

With a total of 1,000 books available to be shipped from Brooklyn, the equation of the number of books shipped from Brooklyn is:

B₁ + B₂ = 1,000

....(i)

And since, a total of 2,000 books are available to be shipped from Queens, the equation of the number of books shipped from Queens is:

Q₁ + Q₂ = 2,000

....(ii)

The total books shipped to each location must be 1,500 each.

The equations for the number of books shipped to each store is:

B₁ + Q₁ = 1,500....(iii)

B₂ + Q₂ = 1,500

....(iv)

The cost to ship a book from Brooklyn to Long Island is, $5.

The cost to ship a book from Brooklyn to Manhattan is, $1.

The cost to ship a book from Queens to Long Island is, $4.

The cost to ship a book from Queens to Manhattan is, $2.

It is provided that the transportation budget is $8,540.

The equation for the total cost of shipping:

5B₁ + B₂ + 4Q₁ + 2Q₂ = 8540

....(v)

The total number of books shipped is 3000. The equation of total number of books shipped is:

B₁ + B₂ + Q₁ + Q₂ = 3000

....(vi)

Consider equation (i):

B₁ + B₂ = 1,000

B₁ = 1000 - B₂....(vii)

Consider equation (iv):

B₂ + Q₂ = 1,500

Q₂ = 1500 - B₂....(viii)

Equate (vii) and (viii) in equation (vi):

B₁ + B₂ + Q₁ + Q₂ = 3000

(1000 - B₂) + B₂ + Q₁ + (1500 - B₂) = 3000

Q₁ - B₂ = 500....(ix)

Equate (vii) and (viii) in equation (v):

5B₁ + B₂ + 4Q₁ + 2Q₂ = 8540

5(1000 - B₂) + B₂ + 4Q₁ + 2(1500 - B₂) = 8540

4Q₁ - 6B₂ = 540....(x)

Solve (ix) and (x) simultaneously:

Q₁ - B₂ = 500 } × 6

4Q₁ - 6B₂ = 540

⇒

6Q₁ - 6₂ = 3000

4Q₁ - 6B₂ = 540

2Q₁= 2460

Q₁ = 1230

Substitute Q₁ = 1230 in (ix) and determine B₂ as follows:

1230 - B₂ = 500

B₂ = 730

Substitute B₂ = 730 in (vii) to determine B₁:

B₁ = 1000 - B₂

= 1000 - 730

= 270

B₁ = 270

Substitute B₂ = 730 in (viii) to determine Q₂:

Q₂ = 1500 - B₂

= 1500 - 730

= 770

Q₂ = 770

Thus, number of books shipped from:

(1) Brooklyn to Long Island = 270

(2) Brooklyn to Manhattan = 730

(3) Queens to Long Island = 1230

(4) Queens to Manhattan = 770