Some years ago it was estimated that the demand for steel approximately satisfied the equation p equals 201 minus 20 x, and the total cost of producing x units of steel was Upper C (x )equals 153 plus 41 x. (The quantity x was measured in millions of tons and the price and total cost were measured in millions of dollars.) Determine the level of production and the corresponding price that maximize the profits.

Question

contestada

Respuesta :

windyyork windyyork · Answer 1 · 2020-03-11T09:23:33+01:00

Answer: The level of production is 4 units and the corresponding price that maximize the profits is $121.

Step-by-step explanation:

Since we have given that

the demand for steel would be

[tex]p=201-20x[/tex]

[tex]R(x)=xp(x)=x(201-20x)=201x-20x^2[/tex]

Total cost of producing x units of steel would be

[tex]C(x)=153+41x[/tex]

Profit = R(x)-C(x)

So, it becomes,

[tex]profit=201x-20x^2-153-41x\\\\P(x)=-20x^2+160x-153[/tex]

First we derivative w.r.t to x we get that

[tex]P'(x)=-40x+160[/tex]

Now, we will find the critical points,

[tex]-40x+160=0\\\\-40x=-160\\\\x=\dfrac{-160}{-40}\\\\x=4[/tex]

So, now second derivative w.r.t. x.

We get that

[tex]P''(x)=-40<0[/tex]

So, there will be maximum profit at 4 units of level of production and the corresponding price that maximize the profit would be

[tex]p(x)=201-20x\\\\p(4)=201-20(4)\\\\p(4)=201-80\\\\p(4)=121[/tex]

Hence, the level of production is 4 units and the corresponding price that maximize the profits is $121.