Question 85175
Income A baker has 150 units of flour, 90 of sugar, and
150 of raisins. A loaf of raisin bread requires 1 unit of flour,
1 of sugar, and 2 of raisins, while a raisin cake needs 5, 2,
and 1 units, respectively.
:
Graphing will be useful here.
:
Let x = no. of loaves of raisin bread
Let y = no. of raisin cakes
:
Flour equation: 
1x + 5y = 150
5y = 150 - x
y = 30 - .2x; green
:
Sugar equation
1x + 2y = 90
2y = 90 - x
y = 45 - .5x; purple
:
Raisin equation
2x + y = 150
y = 150 - 2x; black
:
{{{ graph( 300, 200, -10, 140, -10, 40, 30-.2x,45-.5x,150-2x) }}} 
:
a. If raisin bread sells for $1.75 a loaf and raisin cake for
$4.00 each, how many of each should be baked so that
gross income is maximized?
:
Look at the  corners of the area of feasibility, we have
x = 0, y = 30
x = 50, y = 20
x = 70, y = 10
x = 75, y = 0
: 
b. What is the maximum gross income?
:
Using these value for x and y, find the income for each kind.
x = 50, y = 20 would have the greatest income:
(1.75*50) + (4 * 20) = $167.50
:
c. Does it require all of the available units of flour, sugar,
and raisins to produce the number of loaves of raisin
bread and raisin cakes that produce the maximum profit?
:
No
:
If not, how much of each ingredient is left over?
:
Flour equation:
1(50) + 5(20) = 150, used all the flour
Sugar equation:
1(50) + 2(20) = 90, used all the sugar
Raisin equation
2(50) + 1(20) = 140, 150 - 120 = 30 units of raisins left over
:
I think the method is valid, but check my math here.