A shingle company manufactures premium grade shingles and high grade shingles using two common ingredients: asphalt and gravel. A batch of premium grade shingles requires 19 pounds of asphalt and 9 pounds of gravel and sells for $40 per batch. A batch of?

A shingle company manufactures premium grade shingles and high grade shingles using two common ingredients: asphalt and gravel. A batch of premium grade shingles requires 19 pounds of asphalt and 9 pounds of gravel and sells for $40 per batch. A batch of high grade shingles requires 8 pounds of asphalt and 15 pounds of gravel and sells for $39 per batch. The shingle company has 287 pounds of asphalt and 335 pounds of gravel on hand on a certain day. How many batches of each product should the shingle company make that day to maximize its profit?

If x is the number of batches of premium grade shingles and y
is the number of batches of high grade shingles made that day, answer the following questions connected with the problem:
a) The linear inequality that
x and y must satisfy so as not to exceed the number of pounds of asphalt available is:
b) The linear inequality that
x and y must satisfy to meet the order for ounces of emeralds is:

1 Answer
Cesareo R.
Sep 1, 2017

See below.

Explanation:

Calling

#x_1# = num. of PGS batch produced.
#x_2# = num. of HGS batch produced.
#c_1# = PGS sell price/batch = $40
#c_2# = HGS sell price/batch = $39
#a_(11) = 19# asphalt component for PGS
#a_(12) = 9# gravel component for PGS
#a_(21) = 8# asphalt component for HGS
#a_(22) = 15# gravel component for HGS
#b_1 = 287# asphalt total
#b_2 = 335# gravel total
#X = (x_1,x_2)#

So we have:

Objective function

#f(X)=c_1 x_1+c_2x_2#

Restrictions #Omega#

#((a_(11),a_(12)),(a_(21),a_(22)))((x_1),(x_2)) le ((b_1),(b_2))#

#x_1 ge 0#
#x_2 ge 0#

So the problem is

#max_(X in Omega)f(X)#

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