Math Problem Statement

1st attempt A firm produces output (y ) using two inputs (x1 and x2 ). The two inputs are perfect substitutes, and the firm's production function is

y=4x1+2.00x2 .

Which of the following input combinations (given in (x1,x2) form) is capable of producing 96.00 units (or more)? Sort into the "Yes" column if an input combination is capable of hitting the target and the "No" column if it is not.

Items (7 items) (Drag and drop into the appropriate area below) (24,0)(5,28)(15,20)(17,10)(12,18)(16,24)(0,48)

Solution

To determine if each input combination meets or exceeds the target output of 96 units, we can substitute each pair (x1,x2)(x_1, x_2) into the production function:

y=4x1+2x2y = 4x_1 + 2x_2

We’ll evaluate each pair accordingly and compare the result with 96.

  1. For (24, 0): y=4(24)+2(0)=96+0=96(Yes)y = 4(24) + 2(0) = 96 + 0 = 96 \quad (\text{Yes})

  2. For (5, 28): y=4(5)+2(28)=20+56=76(No)y = 4(5) + 2(28) = 20 + 56 = 76 \quad (\text{No})

  3. For (15, 20): y=4(15)+2(20)=60+40=100(Yes)y = 4(15) + 2(20) = 60 + 40 = 100 \quad (\text{Yes})

  4. For (17, 10): y=4(17)+2(10)=68+20=88(No)y = 4(17) + 2(10) = 68 + 20 = 88 \quad (\text{No})

  5. For (12, 18): y=4(12)+2(18)=48+36=84(No)y = 4(12) + 2(18) = 48 + 36 = 84 \quad (\text{No})

  6. For (16, 24): y=4(16)+2(24)=64+48=112(Yes)y = 4(16) + 2(24) = 64 + 48 = 112 \quad (\text{Yes})

  7. For (0, 48): y=4(0)+2(48)=0+96=96(Yes)y = 4(0) + 2(48) = 0 + 96 = 96 \quad (\text{Yes})

Sorted Combinations

  • Yes (Meets or exceeds 96): (24,0),(15,20),(16,24),(0,48)(24, 0), (15, 20), (16, 24), (0, 48)
  • No (Does not meet 96): (5,28),(17,10),(12,18)(5, 28), (17, 10), (12, 18)

Would you like further details on how these calculations were done, or do you have any other questions?

Additional Questions

  1. How would the required inputs change if the production function changed to y=3x1+2x2y = 3x_1 + 2x_2?
  2. If the target output increased to 120, which combinations would meet or exceed it?
  3. What would happen if one of the inputs doubled in productivity?
  4. How would the analysis change if the inputs were no longer perfect substitutes?
  5. How can we use inequalities to solve for general conditions for meeting output targets?

Tip

In problems with perfect substitutes, maximize output by emphasizing the more productive input unless constraints or other factors dictate otherwise.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Algebra
Production Functions

Formulas

y = 4x1 + 2x2

Theorems

Perfect Substitutes in Production

Suitable Grade Level

Grades 10-12