Question

A manufacturer producing pens has the following information regarding the cost of production of pens:

Output (𝑄)	1	2	3
Total Costs (𝑇𝐶)	4	13	32

"If the total cost function is of the form $𝑇𝐶(𝑄) = 𝑎𝑄^2 + 𝑏𝑄 + 𝑐$ where 𝑎, 𝑏, and 𝑐 are constants, then the value of $𝑇𝐶(𝑄) at 𝑄 = 4$ is (in integer)."

Answer 1

Correct Answer: 61

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:

\[ c = 4 - a - b \]

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

Substituting \( c \) into the second equation:

\[ 13 = 4a + 2b + (4 - a - b) \] \[ 13 = 3a + b + 4 \] \[ 3a + b = 9 \]

Substituting \( c \) into the third equation:

\[ 32 = 9a + 3b + (4 - a - b) \] \[ 32 = 8a + 2b + 4 \] \[ 8a + 2b = 28 \] \[ 4a + b = 14 \]

Step 4: Solving for \( a \), \( b \), and \( c \)

We now have two equations:

• \( 3a + b = 9 \)
• \( 4a + b = 14 \)

Subtracting the equations:

\[ (4a + b) - (3a + b) = 14 - 9 \] \[ a = 5 \]

Substituting \( a = 5 \) into \( 3a + b = 9 \):

\[ 15 + b = 9 \] \[ b = -6 \]

Substituting \( a = 5 \) and \( b = -6 \) into \( c = 4 - a - b \):

\[ c = 4 - 5 - (-6) = 5 \]

Step 5: Finding the Cost Function

Thus, the total cost function is:

\[ T C(Q) = 5Q^2 - 6Q + 5 \]

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

Substituting \( Q = 4 \):

\[ T C(4) = 5(4)^2 - 6(4) + 5 \] \[ = 5(16) - 24 + 5 \] \[ = 80 - 24 + 5 \] \[ = 61 \]

Final Answer:

The total cost at \( Q = 4 \) is 61.