Question

Let​ R(x), C(x), and​ P(x) be,​ respectively, the​ revenue, cost, and​ profit, in​ dollars, from the production and sale of x items. If ​R(x) = 6x and ​C(x) equals = 0.002x^2+2.2x+40​, find each of the following. a. p(x) b.p(100) c. P'(x) d.P'(100)

Answer 1

Answer:

$a.\ P(x) = - 0.002x^2+3.8x-40$

b. 320

$c.\ P'(x) = -0.004 x + 3.8$

d. 3.76

Step-by-step explanation:

Given that:

Revenue function:

$R(x) = 6x$

Cost Function:

$C(x) = 0.002x^2+2.2x+40$

We know that,

Answer a: Profit = Revenue - Cost

$\Rightarrow P(x) = R(x) - C(x)\\\Rightarrow P(x) = 6x - (0.002x^2+2.2x+40)\\\Rightarrow P(x) = - 0.002x^2+3.8x-40$

Answer b:

P(100) = ?

Putting value of x as 100 in above equation:

$P(100) = - 0.002\times 100^2+3.8 \times 100-40\\\Rightarrow P(100) = -20+380 -40\\\Rightarrow P(100) = 320$

Answer c:

P'(x) = ?

Differentiating the equation $P(x) = - 0.002x^2+3.8x-40$

$P'(x) = -2 \times 0.002 x^{2-1} + 3.8 + 0\\P'(x) = -0.004 x + 3.8$

Answer d:

P'(100) = ?

Putting x = 100 in equation $P'(x) = -0.004 x + 3.8$

$P'(100) = -0.004 \times 100 + 3.8\\P'(100) = -0.4 + 3.8\\P'(100) = 3.76$