Question

A manufacturer of handcrafted wine racks has determined that the cost to produce x units per month is given by upper c equals 0.2 x squared plus 10 comma 000.how fast is the cost per month changing when production is changing at the rate of 13units per month and the production level is 70​units

Answer 1

Answer:

The cost per month is increasing at a rate $365.

Explanation:

Differentiation Formula

• $\frac{d}{dx}(x^n)= nx^{n-1}$  
• $\frac{d}{dx}(a)=0$             [ where a is a constant]
• $\frac{d}{dx}(ax^n)=a \frac{d}{dx}(x^n)= anx^{n-1}$

Given that,

A manufacturer of handcrafted wine racks has determined that the cost to produce x units per month is given by

$c=0.2x^2+10,000$.

Again given that,

the rate of changing production is 13 unit per month

i.e $\frac{dx}{dt}=13$

To find the cost per month, we need to find out the value $\frac{dc}{dt}$ when production is changing at the rate 13 units per month and the production is 70 units.

$c=0.2x^2+10,000$

Differentiating with respect to t

$\frac{d}{dt}(c)=\frac{d}{dt}(0.2x^2)+\frac{d}{dx}(10,000)$

$\Rightarrow \frac{dc}{dt}=0.2\frac{d}{dt}(x^2)+\frac{d}{dx}(10,000)$

$\Rightarrow \frac{dc}{dt}=0.2\times 2x^{2-1}\frac{dx}{dt}+0$

$\Rightarrow \frac{dc}{dt}=0.4x\frac{dx}{dt}$

Plugging $\frac{dx}{dt}=13$

$\Rightarrow \frac{dc}{dt}=0.4x\times 13$

$\Rightarrow \frac{dc}{dt}=5.2x$

$\frac{dc}{dt}|_{x=70}=5.2\times 70$ [ plugging x=70]

            =364

[ The unit of c is not given. Assume that the unit of c is dollar.]

The cost per month is increasing at a rate $365.