# Linear Programming Word Problems

A bakery has bought 250 pounds of muffin dough. They want to make waffles or muffins in half-dozen packs out of it. Half a dozen of muffins requires 1 lb of dough and a pack of waffles uses 3/4 lb of dough. It take bakers 6 minutes to make a half-dozen of waffles and 3 minutes to make a half-dozen of muffins. Their profit will be \$1.50 on each pack of waffles and \$2.00 on each pack of muffins. How many of each should they make to maximize profit, if they have just 20 hours to do everything?

I'm confused with Linear Programming word problems -

I first made - ( $x =$ cost of waffles, etc.)

$C(x,y) = 1.5x + 2y$

I need help with the other equations/inequalities. I know how to graph it.

• What word are you having problems with? – user228113 Feb 27 '16 at 13:32
• I can't figure out the other inequalities – user318237 Feb 27 '16 at 13:33
• hint: one constraint should be the time (you will need to convert either minutes to hours, or hours to minutes, your choice), and one constraint will be the amount of dough they have. Your objective function is correct, though it's not in the correct units :) – wz-billings Jun 10 '18 at 22:01

"A bakery has bought 250 pounds of muffin dough. They want to make waffles or muffins in half-dozen packs out of it. Half a dozen of muffins requires 1 lb of dough and a pack of waffles uses 3/4 lb of dough. It take bakers 6 minutes to make a half-dozen of waffles and 3 minutes to make a half-dozen of muffins. Their profit will be $1.50 on each pack of waffles and$2.00 on each pack of muffins. How many of each should they make to maximize profit, if they have just 20 hours to do everything?"

You certainly should NOT "make x= cost of waffles, etc." since that is not what is asked. The question is "How many of each should they make" so you should let x be the number of half-dozen packs of waffles and let y be the number of half-dozen packs of muffins.
You are told "Half a dozen of muffins requires 1 lb of dough and a pack of waffles uses 3/4 lb of dough" so to make x packs of waffles and y packs of muffins requires x+ (3/4)y pounds of dough. You are also told "a bakery has bought 250 pounds of muffin dough. (I am not at all clear how you make waffles out of muffin dough but apparently you can.) You cannot use more dough than you have but you can use less: $x+ (3/4)y\le 250$.

You are told that "It takes bakers 6 minutes to make a half dozen waffles and 3 minutes to make a half dozen of muffins". So it will require 6x+ 3y minutes to make x packs of waffles and y packs of muffins. You are also told that "they have just 20 hours (20 hours*60 minutes/hour= 1200 minutes) to do everything". They cannot use more than 1200 minutes but they can use less: $6x+ 3y\le 1200$.

"Their profit will be 1.50 on each pack of waffles and 2.00 on each pack of muffins" so their profit on x packs of waffles and y packs of muffins is 1.50x+ 2.00y as you say.

The problem is to maximize the profit 1.5x+ 2.0y subject to the constraints $x+ (3/4)y\le 250$ and $6x+ 3y\le 1200$.

• the answer format is - ____,____ or ____,____ packs of muffins and waffles, respectively. I graphed it and found the intersection (desmos.com/calculator/n87ozw9zi2) Is it 100,200. How would i put it in the answer format? – user318237 Feb 27 '16 at 14:15

If we denote the number of batches of muffins as "a" and the number of batches of waffles as "b," we are then supposed to maximize the profit function

P = 2a + 1.5b

subject to the following constraints: a>=0, b>=0, a + (3/4)b <= 250, and 3a + 6b <= 1200. The third constraint can be rewritten as 4a + 3b <= 1000.

Use the simplex method on these coefficients, and you should find the maximum profit to be $500 when a = 250 and b = 0. So, make 250 batches of muffins, no waffles. You use up all the dough, have 450 minutes left, and have$500 profit, the maximum amount.

• Please, use MathJax (i.e. LaTeX commands) for mathematical notations. – Taroccoesbrocco Jun 10 '18 at 21:44