Question

A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. If x machines are made, then the unit cost is given by the function
C (x) = 0.8x ^ 2 - 256x +25,939 . How many machines must be made to minimize the unit cost? Do not round your answer.
Number of copy machines:

Answer 1

Answer:

Number of copy machines must be made to minimize the unit cost=160.

Step-by-step explanation:

We are given that the unit cost function C ( the cost in dollars to make each copy machine)

If machines are made =x

Then the unit cost function is given by

$C(x)=0.8x^2-256x+25939$

We have to find the number of copy machines for  minimize the unit  cost

$C(x)=0.8x^2-256x+25939$

Differentiate with respect to x

Then we get

$\frac{\mathrm{d}C}{\mathrm{d}x}=1.6x-256$ ......(equation I)

To find the value of x then we susbtitute $\frac{\mathrm{d}C}{\mathrm{d}x}$ is equal to zero

$\frac{\mathrm{d}C}{\mathrm{d}x}=0$

$1.6x-256=0$

$1.6x=256$

$x=\frac{256}{1.6}$

By using division property of equality

$x= 160$

Again differentiate the equation I with respect to x then we get

$\frac{\mathrm{d}^2 C}{\mathrm{d}^2 x}=1.6 >0$

Hence, the unit cost is minimize for x=160

Therefore, the number of copy machines must be made to minimize the unit cost =160

Answer 2

About 169 machines are needed to minimize the unit cost to $5459.

Cost

The cost of a goods is the total amount of money spent in the production of the goods.

Given the cost equation:

C(x) = 0.8x² - 256x +25,939

The minimum cost is at C'(x) = 0, hence:

C'(x) = 1.6x - 256

1.6x - 256 = 0

x = 160

C(160) = 0.8(160)² - 256(160) + 25939 = 5459

About 169 machines are needed to minimize the unit cost to $5459.

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