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solniwko [45]
3 years ago
5

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
Business
1 answer:
Fantom [35]3 years ago
4 0

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.

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