Question

DuocUC 2) The cost C, in pesos, for the production of x meters of a certain fabric can be calculated through the function: (x+185) C(x)=81300-6x+ 20000 a) It is known that C(90) 5.344. Interpret this result. (2 points) b) Calculate C'(x) (2 points) 3 x²+111x-0.87 20000 2000 c) Function C calculates the cost while producing a maximum of 500 meters of fabric. Determine the values of x at which the cost of production is increasing and the values of x at which the cost is decreasing. (3 points) d) If a maximum of 500 meters of fabric are produced, what is the minimum production cost? (

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a) To interpret the result C(90) = 5,344, we substitute x=90 into the equation C(x) = 81300 - 6x + 20000:

C(90) = 81300 - 6(90) + 20000

= 81300 - 540 + 20000

= 81800 - 540

= 81260

The result C(90) = 5,344 means that the cost for producing 90 meters of fabric is 5,344 pesos.

Answer: The cost for producing 90 meters of fabric is 5,344 pesos.

b) To calculate C'(x), we differentiate the function C(x) = 81300 - 6x + 20000 with respect to x:

C'(x) = -6

Answer: C'(x) = -6

c) To determine the values of x at which the cost of production is increasing or decreasing for C(x) with a maximum of 500 meters of fabric, we need to analyze the sign of C'(x).

Since C'(x) is constant and equal to -6, it is always negative. Therefore, the cost of production is always decreasing for C(x) with a maximum of 500 meters of fabric.

Answer: The cost of production is always decreasing for C(x) with a maximum of 500 meters of fabric.

d) To find the minimum production cost when a maximum of 500 meters of fabric is produced, we substitute x=500 into the equation C(x) = 81300 - 6x + 20000:

C(500) = 81300 - 6(500) + 20000

= 81300 - 3000 + 20000

= 98300 - 3000

= 95300

The minimum production cost when a maximum of 500 meters of fabric is produced is 95,300 pesos.

Answer: The minimum production cost when a maximum of 500 meters of fabric is produced is 95,300 pesos.

C(90) = 81300 - 6(90) + 20000

= 81300 - 540 + 20000

= 81800 - 540

= 81260

The result C(90) = 5,344 means that the cost for producing 90 meters of fabric is 5,344 pesos.

Answer: The cost for producing 90 meters of fabric is 5,344 pesos.

b) To calculate C'(x), we differentiate the function C(x) = 81300 - 6x + 20000 with respect to x:

C'(x) = -6

Answer: C'(x) = -6

c) To determine the values of x at which the cost of production is increasing or decreasing for C(x) with a maximum of 500 meters of fabric, we need to analyze the sign of C'(x).

Since C'(x) is constant and equal to -6, it is always negative. Therefore, the cost of production is always decreasing for C(x) with a maximum of 500 meters of fabric.

Answer: The cost of production is always decreasing for C(x) with a maximum of 500 meters of fabric.

d) To find the minimum production cost when a maximum of 500 meters of fabric is produced, we substitute x=500 into the equation C(x) = 81300 - 6x + 20000:

C(500) = 81300 - 6(500) + 20000

= 81300 - 3000 + 20000

= 98300 - 3000

= 95300

The minimum production cost when a maximum of 500 meters of fabric is produced is 95,300 pesos.

Answer: The minimum production cost when a maximum of 500 meters of fabric is produced is 95,300 pesos.

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