A sausage company makes two different kinds of hot dogs, regular and all beef. Each pound of all-beef hot dogs requires 0.75 lb of beef and 0.2 lb of spices, and each pound of regular hot dogs requires 0.18 lb of beef, 0.3 lb of pork, and 0.2 lb of spices. Suppliers can deliver at most 1,020 lb of beef, at most 600 lb of pork, and at least 500 lb of spices. If the profit is $0.70 on each pound of all-beef hot dogs and $0.40 on each pound of regular hot dogs, how many pounds of each should be produced to obtain maximum profit? What is the maximum profit? Round your profit to the nearest cent, another answers - to the nearest whole number.

A)Maximum profit of $1150.00 with 2000 lbs of regular and 500 lbs of all beef.
B)Maximum profit of $1416.00 with 2000 lbs of regular and 880 lbs of all beef.
C)Maximum profit of $1752.00 with 880 lbs of regular and 2000 lbs of all beef.
D)Maximum profit of $1300.00 with 1500 lbs of regular and 1000 lbs of all beef.
E)Maximum profit of $1450.00 with 1000 lbs of regular and 1500 lbs of all beef.

Question

A sausage company makes two different kinds of hot dogs, regular and all beef. Each pound of all-beef hot dogs requires 0.75 lb of beef and 0.2 lb of spices, and each pound of regular hot dogs requires 0.18 lb of beef, 0.3 lb of pork, and 0.2 lb of spices. Suppliers can deliver at most 1,020 lb of beef, at most 600 lb of pork, and at least 500 lb of spices. If the profit is $0.70 on each pound of all-beef hot dogs and $0.40 on each pound of regular hot dogs, how many pounds of each should be produced to obtain maximum profit? What is the maximum profit? Round your profit to the nearest cent, another answers - to the nearest whole number.
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A)Maximum profit of $1150.00 with 2000 lbs of regular and 500 lbs of all beef.
B)Maximum profit of $1416.00 with 2000 lbs of regular and 880 lbs of all beef.
C)Maximum profit of $1752.00 with 880 lbs of regular and 2000 lbs of all beef.
D)Maximum profit of $1300.00 with 1500 lbs of regular and 1000 lbs of all beef.
E)Maximum profit of $1450.00 with 1000 lbs of regular and 1500 lbs of all beef.

Quizplus · Accepted Answer

Let x be the number of pounds of all-beef hot dogs and y be the number of pounds of regular hot dogs. Then the objective is to maximize the profit function:

Profit = 0.70x + 0.40y

subject to the constraints:

0.75x + 0.18y ≤ 1020 (beef constraint)
0.3y ≤ 600 (pork constraint)
0.2x + 0.2y ≥ 500 (spice constraint)
x ≥ 0, y ≥ 0 (non-negativity constraints)

Using these constraints, we can solve for x and y to obtain the maximum profit. One way to do this is to use a linear programming solver or graph the constraints to find the feasible region and then test the vertices of the region for the maximum. Another way is to use the simplex method, which is a systematic procedure for solving linear programming problems.

Using the simplex method, we can write the constraints in standard form:

0.75x + 0.18y + s1 = 1020
0.3y + s2 = 600
-0.2x - 0.2y + s3 = -500

where s1, s2, and s3 are slack variables introduced to convert the inequalities to equations.

Then we can write the objective function as:

-0.70x - 0.40y + z = 0

where z is the value of the objective function.

Putting these equations into a simplex tableau, we can solve for the values of x, y, and z that maximize the objective function. The optimal solution is:

x = 880, y = 2000, z = 1416

Therefore, the maximum profit is $1416.00 and the optimal production plan is to produce 880 pounds of all-beef hot dogs and 2000 pounds of regular hot dogs.

A Sausage Company Makes Two Different Kinds of Hot Dogs