Nicole decides to make two sizes of bottles of BBQ sauce, 16 ounces and 8 ounces. Her total production is 1,280 ounces for the summer. She has room for up to 100 bottles in her booth at the farmers market and wants to maximize her income from their sale. Let x represent the number of 16-oz bottles sold and y represent the number of 8-oz bottles. The 16-oz bottles sell for $25 and the 8-oz bottles sell for $15.

Step-by-step explanation:

Given that Nicole decides to make two sizes of bottles of BBQ sauce, 16 ounces and 8 ounces. Her total production is 1,280 ounces for the summer. She has room for up to 100 bottles in her booth at the farmers market and wants to maximize her income from their sale

Let x be the 16 oz bottle and y the 8 oz bottle

Profit Z = 25x+15y and objective is to maximize Z

constrains are

Solving we get point of intersectin is x = 60 and y =40

Other corner points are x = 80, y=0 or x=0, y =100

Profit for (60,40) is 25(60)+15(40) = 2100

Profit for (80,0) is 25(80) = 2000

Profit for (0,100) is 1500

So maximum profit at 60, 40 for 16 oz and 8 oz

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

no problobm

step-by-step explanation: