Question

PLEASE HELPPPP

The cost function in a computer manufacturing plant is C(x) = 0.28x^2-0.7x+1, where C(x) is the cost per hour in millions of dollars and x is the number of items produced per hour in thousands. Determine the minimum production cost.

Answer 1

The minimum cost is 0.5625.

Explanation:

The cost function is

C(x) = 0.28x^2 - 0.7 x + 1

Differentiate with respect to x.

`C = 0.28x^2 - 0.7 x + 1\\\\(dC)/(dt) = 0.56 x - 0.7\\\\(dC)/(dt) = 0\\\\0.56 x - 0.7 = 0\\\\x = 1.25`

The minimum value is

c = 0.28 x 1.25 x 1.25 - 0.7 x 1.25 + 1

C = 0.4375 - 0.875 + 1

C = 0.5625

Answer 2

Given:

The cost function is:

`C(x)=0.28x^2-0.7x+1`

where C(x) is the cost per hour in millions of dollars and x is the number of items produced per hour in thousands.

To find:

The minimum production cost.

Solution:

We have,

`C(x)=0.28x^2-0.7x+1`

It is a quadratic function with positive leading efficient. It means it is an upward parabola and its vertex is the point of minima.

If a quadratic function is
`f(x)=ax^2+bx+c`, then the vertex of the parabola is:

`\text{Vertex}=\left(-(b)/(2a),f(-(b)/(2a))\right)`

In the given function,
`a=0.28, b=-0.7, c=1`. So,

`-(b)/(2a)=-(-0.7)/(2(0.28))`

`-(b)/(2a)=1.25`

Putting
`x=1.25` in the given function to find the minimum production cost.

`C(x)=0.28(1.25)^2-0.7(1.25)+1`

`C(x)=0.28(1.5625)-0.875+1`

`C(x)=0.4375+0.125`

`C(x)=0.5625`

Therefore, the minimum production cost is 0.5625 million dollars.