contestada

The cost (in dollars) of producing x units of a certain commodity is C(x) = 7000 + 6x + 0.15x2. (a) Find the average rate of change of C with respect to x when the production level is changed from x = 100 to the given value. (Round your answers to the nearest cent.) (i) x = 103 $ per unit (ii) x = 101 $ per unit (b) Find the instantaneous rate of change of C with respect to x when x = 100. (This is called the marginal cost.) $ per unit

Respuesta :

Answer:

The instantaneous rate is 36.

Step-by-step explanation:

Given the cost of producing a commodity,

[tex]C(x) = 7000 + 6x + 0.15x^{2}[/tex]

Now calculate the average rate of change in cost of producing commodity. Ā 

Use below formula to calculate average rate of change when (i) x = 103 $ per unit:

[tex]\text{Average rate of change} = \frac{C(103) ā€“ C(100) }{103 - 100} \\C(x) = 7000 + 6x + 0.15x^{2} \\C(100) = 7000 + 6 \times 100 + 0.15 (100)^{2} = 9100 \\C(103) = 7000 + 6 \times 103 + 0.15 (103)^{2} = 9209.35 \\\text{Average rate of change} = \frac{9209.35 - 9100}{103 - 100} = 36.45[/tex]

Now calculate average rate of change when (ii) x = 101 $ per unit:

[tex]\text{Average rate of change} = \frac{C(101) ā€“ C(100) }{101 - 100} \\C(x) = 7000 + 6x + 0.15x^{2} \\C(100) = 7000 + 6 \times 100 + 0.15 (100)^{2} = 9100 \\C(101) = 7000 + 6 \times 101 + 0.15 (101)^{2} = 9136.15 \\\text{Average rate of change} = \frac{9136.15 - 9100}{101 - 100} = 36.15[/tex]

Now to find the Instantaneous Rate of Change we just need to find the differentiation of given function.

[tex]C(x) = 7000 + 6x + 0.15x^{2} \\Cā€™(x) = 6 + 0.3x \\[/tex]

Instantaneous Rate of Change when x = 100

[tex]= 6 + 0.3 \times (100) \\= 36[/tex]

Otras preguntas