A banquet hall offers two types of tables for rent: 6-person rectangular tables at a cost of $21 each and 10-person round tables at a cost of $54 each. Kelly would like to rent the hall for a wedding banquet and needs tables for 230 people. The room can have a maximum of 33 tables and the hall only has 15 rectangular tables available. How many of each type of table should be rented to minimize cost and what is the minimum cost?

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Let C be the cost. x be the number of rectangular tables and y be the number of round tables.

Thus

C=21x+54y

s.t.

6x+10y `>=230` ( no. of peoples )

`x<=15` ( available restriction)

`x+y<=33` (space restriction)

`x,y>=0`

Thus from our problem

Min`C=21x+54y`

s.t.

`x<=15`

`3x+5y<=115`

`x+y<=33`

`x,y>=0`

Let draw graph of constrains

Red: `3x+5y>=115`

green : `x+y<=33`

let red meet y-axis at P(0,23)

green meets y axis Q(0,33)

red and x`<=15` meet at R (15,14)

green and `x<=15` meet at S(15,18)

The cost C at P=21 x 0+54 x23= 1242

The cost C at Q= 21 x 0 + 54 x 33= 1782

The cost C at R=21 x15+54 x14=1071

The cost C at S=21 x 15+ 54 x 18=1287

Min cost( 1242,1762,1071,1287)=1071 at R(15,14).

**Ans. minmum cost is 1071 $ at rectangular tables are 15 ad round tables are 14.**

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