Question:

A manufacturer producing pens has the following information regarding the cost of production of pens:
Output (𝑄)123
Total Costs (𝑇𝐢)41332

"If the total cost function is of the form 𝑇𝐢(𝑄)=π‘Žπ‘„2+𝑏𝑄+𝑐𝑇𝐢(𝑄) = π‘Žπ‘„^2 + 𝑏𝑄 + 𝑐 where π‘Ž, 𝑏, and 𝑐 are constants, then the value of 𝑇𝐢(𝑄)at𝑄=4𝑇𝐢(𝑄) at 𝑄 = 4 is (in integer)."

Updated On: Feb 10, 2025
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Correct Answer: 61

Solution and Explanation

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 4 = a(1)^2 + b(1) + c 13=a(2)2+b(2)+c 13 = a(2)^2 + b(2) + c 32=a(3)2+b(3)+c 32 = a(3)^2 + b(3) + c

Step 2: Expanding the Equations

Expanding the equations:

4=a+b+c 4 = a + b + c 13=4a+2b+c 13 = 4a + 2b + c 32=9a+3b+c 32 = 9a + 3b + c

From the first equation:

c=4βˆ’aβˆ’b c = 4 - a - b

Step 3: Substituting c c into Equations (2) and (3)

Substituting c c into the second equation:

13=4a+2b+(4βˆ’aβˆ’b) 13 = 4a + 2b + (4 - a - b) 13=3a+b+4 13 = 3a + b + 4 3a+b=9 3a + b = 9

Substituting c c into the third equation:

32=9a+3b+(4βˆ’aβˆ’b) 32 = 9a + 3b + (4 - a - b) 32=8a+2b+4 32 = 8a + 2b + 4 8a+2b=28 8a + 2b = 28 4a+b=14 4a + b = 14

Step 4: Solving for a a , b b , and c c

We now have two equations:

  • 3a+b=9 3a + b = 9
  • 4a+b=14 4a + b = 14

Subtracting the equations:

(4a+b)βˆ’(3a+b)=14βˆ’9 (4a + b) - (3a + b) = 14 - 9 a=5 a = 5

Substituting a=5 a = 5 into 3a+b=9 3a + b = 9 :

15+b=9 15 + b = 9 b=βˆ’6 b = -6

Substituting a=5 a = 5 and b=βˆ’6 b = -6 into c=4βˆ’aβˆ’b c = 4 - a - b :

c=4βˆ’5βˆ’(βˆ’6)=5 c = 4 - 5 - (-6) = 5

Step 5: Finding the Cost Function

Thus, the total cost function is:

TC(Q)=5Q2βˆ’6Q+5 T C(Q) = 5Q^2 - 6Q + 5

Step 6: Calculating TC(4) T C(4)

Substituting Q=4 Q = 4 :

TC(4)=5(4)2βˆ’6(4)+5 T C(4) = 5(4)^2 - 6(4) + 5 =5(16)βˆ’24+5 = 5(16) - 24 + 5 =80βˆ’24+5 = 80 - 24 + 5 =61 = 61

Final Answer:

The total cost at Q=4 Q = 4 is 61.

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