math problem

Question

You own a catering business that makes specialty cakes. Your company has decided to create three types of cakes. To create these cakes, it takes a team that consists of a decorator, a baker, and a design consultant.

Cake A takes the decorator 9 hours, the baker 6 hours, and the design consultant 1 hour to complete.
Cake B takes the decorator 10 hours, the baker 4 hours, and the design consultant 2 hours.
Cake C takes the decorator 12 hours, the baker 4 hour, and the design consultant 1 hour.
Without hiring additional employees, there are 398 decorator hours available, 164 baker hours available, and 58 design consultant hours available. How many of each type of cake can be created?

Use the following guidelines for your answer:

A = The number of cake A that can be produced.
B = The number of cake B that can be produced.
C = The number of cake C that can be produced.

Answer 1

A = number of Cake A

B = number of Cake B

C = number of Cake C

1. Write the system of equations

Decorator hours:

9A+10B+12C = 398

Baker hours:

6A+4B+4C = 164

Design consultant hours:

A+2B+C = 58

1. Solve the system

Isolate A from the designer equation,

A = 58-2B-C

Substitute A into the baker equation, simplify and solve for C in terms of B

6(58-2B-C) +4B+4C= 164

348-12B-6C+4B+4C= 164

- 8B - 2C= -184

C = 92-4B

Now substitute A and C into the designer equation,

A= 58-2B-C

A=58-2B-(92-4B)

A=2B-34

Now substitute A and C into

9A +10B +12C = 398

9(2B-34) +10B + 12(92-4B)=398

18B-306 +10B +1104 -48B =398

combine like terms to simplify,

-20B =398 - 798

-20B = -400

B= 20

find A and C,

A= 2B -34

A=2(20) - 34= 6

A= 6

and

C= 92-4B

C= 92- 4(20)=12

C = 12

Final answer: A= 6 cakes B=20 cakes C=12 cakes