Mrs white wants to crochet beach hits and baby afghans for a church fund raising bazaar. she needs 7 hours to make a hat and 4 hours to make an afghan and she has 68 hours available. she wants to make no more than 14 items and no more than 11 afghans. the bazaar will sell hats for $21 each and the afphans for $9 each. how many of each should she make to maximize the income for the bazaar?

This is a maximization problem so we apply derivatives here to determine the unknown variables in the problem. 

In this problem, we represent x as the number of hats made, and y as the number of Afghans made by Mrs White. 
Equation 1 relating to the time it takes to make these is expressed:

7x + 4y = 68
Another equation that represents inequality to the number of hats and Afghans respectively is expressed: 
x<= 14
y<=11

the third equation expresses the income from selling these items expressed as 
P = 21 x + 9y 
we subtitute 1 to 3

P = 9(68-7x)/4 + 21x = 153-15.75x + 21 x = 153 -5.25x 

So by trial and errror, x and y should be integers, we get two cases of which x and y should be

1) x = 4 ; y = 10
2) x = 8 ; y = 3

Subsituting to 3,  P1 = 174$ while P2is equal to $195, Answer then is 8 hats and 3 Afghans in total.

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