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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?

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willlockwood9

THE PROBLEM:

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?

Assume that ALL working hours need to be used while producing A, B, and C quantities simultaneously (although this was not explicitly stated).

If so - the problem is reduced to simple linear algebra…

A*9 + B*10 + C*12 = 398

A*6 + B*4 + C*4 = 164

A*1 + B*2 + C* 1 = 58

The problem can be formulated as M x X = F, where:

Matrix “M” = {[9,10,12][6,4,4][1,2,1]}

Matrix “X” = [A,B,C] (unknown cake quantities)

Matrix “F” = [398,164,58]

And the solution is:

X = M^-1 x F, where

Matrix “M^-1” = {[-0.1,0.35,-0.2][-0.05,-0.075,0.9][0.2,-0.2,-0.6]} (inverse of Matrix “M”)

Performing the matrix multiplication, the result is:

X = [6,20,12] (cakes A, B and C respectively)

you can get the same answer using simple substitution, e.g. calculate ‘C’ from Eq. 3 as: C = 58 - A -2*B, substitute this for ‘C’ in Eq.1 and 2, then calculate ‘B’ from Eq.2 in similar manner, substitute the expression for ‘B’ in Eq. 1 - and you have your answer.

The total number of Cake A is 6, the total number of Cake B is 20 and the total number of Cake C is 12 and this can be determined by forming the linear equations.

Given :

  • 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 hours, 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.

The linear equation can be formed in order to determine the total amount of each type of cake that can be created.

The linear equation represents the total number of hours required by a baker is:

9A + 10B +12C = 398 --- (1)

The linear equation represents the total number of hours required by a decorator is:

6A + 4B + 4C = 164  --- (2)

The linear equation represents the total number of hours required by a design consultant is:

A + 2B + C = 58  --- (3)

Simplify the above system of equations in order to determine the values of A, B, and C.

A = 6 cakes

B = 20 cakes

C = 12 cakes

For more information, refer to the link given below:

https://brainly.com/question/11897796

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