x = number of cups the potter makesy = number of plates the potter makes

x = number of cups the potter makesy = number of plates the potter makes

P = ($2/cup)(x cups) + ($1.50/plate)(y plates),so P = $2x + $1.50y

Time constraint:time on cups + time on plates ≤ time available

x = number of cups the potter makesy = number of plates the potter makes

P = ($2/cup)(x cups) + ($1.50/plate)(y plates),so P = $2x + $1.50y

Time constraint:time on cups + time on plates ≤ time available(6 min./cup)(x cups) + (3min./plate)(y plates) ≤(20 hrs.)(60 min./hr.) or

6x + 3y ≤ 1200Clay constraint:clay for cups + clay for plates ≤ clay available

x = number of cups the potter makesy = number of plates the potter makes

P = ($2/cup)(x cups) + ($1.50/plate)(y plates),so P = $2x + $1.50y

Time constraint:time on cups + time on plates ≤ time available(6 min./cup)(x cups) + (3min./plate)(y plates) ≤(20 hrs.)(60 min./hr.) or

6x + 3y ≤ 1200Clay constraint:clay for cups + clay for plates ≤ clay available(3/4 lb. of clay/cup)(x cups) +(1 lb. of clay/plate)(y plates) ≤ 250 lbs. of clay

.75x + y ≤ 250

x = number of cups the potter makesy = number of plates the potter makes

P = ($2/cup)(x cups) + ($1.50/plate)(y plates),so P = $2x + $1.50y

Time constraint:time on cups + time on plates ≤ time available(6 min./cup)(x cups) + (3min./plate)(y plates) ≤(20 hrs.)(60 min./hr.) or

6x + 3y ≤ 1200Clay constraint:clay for cups + clay for plates ≤ clay available(3/4 lb. of clay/cup)(x cups) +(1 lb. of clay/plate)(y plates) ≤ 250 lbs. of clay

.75x + y ≤ 250Non-negative constraints: x ≥ 0, y ≥ 0

Summary:x = number of cups the potter makesy = number of plates the potter makes

The potter wants to maximize profitP = $2x + $1.50y

Constraints:6x + 3y ≤ 1200.75x + y ≤ 250

x ≥ 0y ≥ 0

400

x

yy ≤ -2x + 400

y ≤ -3/4x +250x ≥ 0y ≥ 0

100

200

300

500

(0, 250)

(1000/3, 0)(0, 0)

(?, ?)

Change from: To: 6x + 3y ≤ 1200 6x + 3y = 1200 .75x + y ≤ 250 ¾ x + y = 250

Solving by Substitution

6x + 3y = 1200¾x + y = 250 y = -¾x +250

6x + 3( -¾x +250) = 1200 6x + (- 9/4)x + 750 = 1200 (15/4)x = 450 15x = 1800 x = 120 6(120) + 3y = 1200

720 + 3y = 1200 3y = 480

y = 160

Change from: To: 6x + 3y ≤ 1200 6x + 3y = 1200 .75x + y ≤ 250 ¾ x + y = 250

Solving by Elimination

6x + 3y = 1200-3(¾x + y = 250)

6x + 3y = 1200 (-9/4)x – 3y = -750 (15/4)x + 0 = 450

(15/4)x = 450 x = 120 6(120) + 3y = 1200

720 + 3y = 1200 3y = 480

y = 160

400

x

yy ≤ -2x + 400

y ≤ -3/4x +250x ≥ 0y ≥ 0

100

200

300

500

(0, 250)

(1000/3, 0)(0, 0)

(120,160)

Corner Points(0, 0) : P = $2(0) + $1.50(0) = $0 + $0 = $0 P = $0 If zero cups and zero plates are made the profit will be $0.

(0, 250) : P = $2(0) + $1.50(250) = $0 + $375 P = $375 If zero cups and 250 plates are made the profit will be $375.

(1000/3, 0) : P = $2(333) + $1.50(0) = $666 + $0 = $666 P = $666If 333 cups are made and 0 plates are made then the profit will be $666.

(120, 160) : P = $2(120) + $1.50(160) = $240 + $240 P = $480 If 120 cups are made and 160 plates are made then the profit will be $480.

For this problem the maximum should be found because we want the maximum profit.

Maximum Profit!!