Finding the Total Cost Function
Step 1: Given System of Equations
Using the data points, we have the following system of equations for the cost function:
4=a(1)2+b(1)+c 13=a(2)2+b(2)+c 32=a(3)2+b(3)+c
Step 2: Expanding the Equations
Expanding the equations:
4=a+b+c 13=4a+2b+c 32=9a+3b+c
From the first equation:
c=4βaβb
Step 3: Substituting c into Equations (2) and (3)
Substituting c into the second equation:
13=4a+2b+(4βaβb) 13=3a+b+4 3a+b=9
Substituting c into the third equation:
32=9a+3b+(4βaβb) 32=8a+2b+4 8a+2b=28 4a+b=14
Step 4: Solving for a, b, and c
We now have two equations:
- 3a+b=9
- 4a+b=14
Subtracting the equations:
(4a+b)β(3a+b)=14β9 a=5
Substituting a=5 into 3a+b=9:
15+b=9 b=β6
Substituting a=5 and b=β6 into c=4βaβb:
c=4β5β(β6)=5
Step 5: Finding the Cost Function
Thus, the total cost function is:
TC(Q)=5Q2β6Q+5
Step 6: Calculating TC(4)
Substituting Q=4:
TC(4)=5(4)2β6(4)+5 =5(16)β24+5 =80β24+5 =61
Final Answer:
The total cost at Q=4 is 61.