 If output is produced according to 𝑌 = 𝑓(𝐿,𝐾) = 4𝐿𝐾, the price of K is £10, and the price of L is £40, 
then the cost minimizing cost minimizing combination of K and L capable of producing 64 units of 
output is
a. 𝐿 = 16 𝑎𝑛𝑑 𝐾 = 1.
b. 𝐿 = 2 𝑎𝑛𝑑 𝐾 = 8.
c. 𝐿 = 2 𝑎𝑛𝑑 𝐾 = 2.
d. 𝐿 = 32 𝑎𝑛𝑑 𝐾 = 32.


The production function is given by \( Y = f(L, K) = 4LK \), where \( Y \) is the output, \( L \) is labor, and \( K \) is capital. The goal is to minimize the cost while producing 64 units of output, i.e., \( Y = 64 \).

### Step 1: Find the constraint based on the output
Since \( Y = 4LK \), we can set \( Y = 64 \) to find the constraint for \( L \) and \( K \):
\[
64 = 4LK
\]
\[
LK = 16
\]
Thus, the product of \( L \) and \( K \) must equal 16.

### Step 2: Set up the cost function
The cost function is the total cost of labor and capital. Given that the price of \( L \) (labor) is £40 and the price of \( K \) (capital) is £10, the total cost \( C \) is:
\[
C = 40L + 10K
\]
We need to minimize this cost subject to the constraint \( LK = 16 \).

### Step 3: Test the options

#### Option a: \( L = 16 \) and \( K = 1 \)
- \( LK = 16 \times 1 = 16 \) (satisfies the constraint)
- Cost: \( C = 40(16) + 10(1) = 640 + 10 = 650 \)

#### Option b: \( L = 2 \) and \( K = 8 \)
- \( LK = 2 \times 8 = 16 \) (satisfies the constraint)
- Cost: \( C = 40(2) + 10(8) = 80 + 80 = 160 \)

#### Option c: \( L = 2 \) and \( K = 2 \)
- \( LK = 2 \times 2 = 4 \) (does not satisfy the constraint)

#### Option d: \( L = 32 \) and \( K = 32 \)
- \( LK = 32 \times 32 = 1024 \) (does not satisfy the constraint)

### Conclusion:
The cost-minimizing combination that satisfies the constraint is **Option b: \( L = 2 \) and \( K = 8 \)**, with a cost of £160.

Would you like more details or have any questions? Here are some relative questions for further practice:

1. How do you derive the cost function in production problems?
2. What is the method to minimize cost when given a production function?
3. How would the cost change if the price of \( K \) were £20 instead of £10?
4. What is the importance of the marginal product of labor and capital in production?
5. How does changing the production function affect the optimal labor-capital combination?

**Tip**: Always verify that the solution satisfies the production constraint before checking the cost.

If output is produced according to Y = f(L,K) = 4LK, the price of K is £10, and the price of L is £40, then the cost minimizing combination of K and L capable of producing 64 units of output is:
a. L = 16 and K = 1.
b. L = 2 and K = 8.
c. L = 2 and K = 2.
d. L = 32 and K = 32.

Solve the problem of minimizing cost with the production function Y = 4LK, where labor costs £40 and capital costs £10. Discover the optimal combination of labor and capital that produces 64 units of output, applying algebraic methods to find the solution. Suitable for high school students in Grades 10-12.

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