Question

A retired couple supplement their income by making fruit pies, which they sell to a local grocery store. During the month of September, they produce apple and grape pies. The apple pies are sold for $1.40 to the grocer, and the grape pies are sold for $1.25. The couple is able to sell all of the pies they produce owing to their high quality. They use fresh ingredients. Flour and sugar are purchased once each month. For the month of September, they have 1,150 cups of sugar and 1,800 cups of flour. Each apple pie requires 1½ cups of sugar and 3 cups of flour, and each grape pie requires 2 cups of sugar and 3 cups of flour.

a. Determine the number of grape and the number of apple pies that will maximize revenues if the couple working together can make an apple pie in 8 minutes and a grape pie in 2 minutes. Together, they plan to work no more than 45 hours. (Round your answers to nearest whole number. Omit the "$" sign in your response.)

Apple	____	Pieces
Grape	_____	Pieces
Revenue	$____

b. Determine the amounts of sugar, flour, and time that will be unused. (Leave no cells blank - be certain to enter "0" wherever required. Round your intermediate calculations and final answers to the nearest whole number.)

Sugar____		cups
Flour_____		cups
Time_____		minutes

Answer 1

Here, let me consider Number of Apple Pies=X1

Number of Grape Pies=X2

They want to Maximize the sales Revenue, hence, Maximize Objective function=1.40*X1+1.25*X2

Constraints:

Sugar: 1.5*X1+2*X2=<1150

Flour: 3*X1+3*X2=<1800

8*X1+2*X2=<2700

X1 and X2>=0

Here, I will use graphical method to calculate the value of X1 and X2

1400 13日 12日 119日 1866 966 | B6日 766 666 500 40-F 300 26日 10日 0 14 28日 426 566 FB 6 21日 35 496 63日 TFB

Point o A B X coordinate (X1) Y coordinate (X2) Value of the objetive function (Z) 0 0 0 575 2875/4 2300/3 0 3220/3 100 500 3100/13 5150/13 21555/26 0 600 750 600 0 840 с 765 D E F G 250 350 1575/2 IT 1350 3375/2 I 675/2 0 945/2

Number of Apple Pies=250 pieces

Number of Grape Pies=350 pieces

Maximize the sales Revenue =1575/2=$787.5

Sugar=1.5*X1+2*X2=(1.5*250)+(2*350)=1075 cups

Flour=3*X1+3*X2=(3*250)+(3*350)=1800 Cups

Time=8*X1+2*X2=(8*250)+(2*350)=2700 Minutes