Question

Suppose that the supply of x units of a product at price p dollars per unit is given by the following.

Answer 1

Answer:

(a) dp/dx = 90/(x+2)

(b) $2.25

(c) approx = $2.25

    exact = $2.22

Step-by-step explanation:

$p = 20 + 90 \ln(2x + 4)$

(a) To find the rate of change, differentiate $p$ with respect to $x$ :

$\implies \frac{d}{dx}(p) = \frac{d}{dx}(20) + \frac{d}{dx}(90 \ln(2x + 4))$

$\implies \frac{dp}{dx}=0+90\times\dfrac{1}{2x+4}\times2$

$\implies \frac{dp}{dx}=\dfrac{180}{2x+4}$

$\implies \frac{dp}{dx}=\dfrac{90}{x+2}$

(b)  Find  $\frac{dp}{dx}$  when $x=38$ :

$\implies \frac{dp}{dx}=\dfrac{90}{38+2}$

$\implies \frac{dp}{dx}=\dfrac{90}{40}$

$\implies \frac{dp}{dx}=2.25$

(c)  Approximate price increase associated with the number of units supplied changing from 38 to 39 is  $\frac{dp}{dx}$  when $x=38$

$\implies \ 2.25$

Exact price increase:

$\implies \left. p \right_{x=39}-\left. p \right_{x=38}\\\\\implies [20 + 90 \ln(2 \times 39 + 4)]-[20 + 90 \ln(2\times38 + 4)]\\\\\implies 416.6047323...-414.3823972...\\\\\implies 2.222335133...\\\\\implies \ 2.22$