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.