1. ## Optimization

1. The company's cost C(x)=x^2-3x+64 dollars to produce x items The selling price p when x hundred units are produced is p(x)=1/4(44-x). Determine the level of production (# of items produced) that maximizes profit.
Here are my steps:

R(x)= x(1/4)(44-x)=11x-(1/4)x^2
R(x)=C(x)
11x-(1/4)^2=x^2-3x+64
derive: 11-(1/2)(x)=2x-3
-(1/2)x-2x=-14
-5/2(x)=-14
5x=28
x=28/5=5.6
Since Profit is P(x)= R(x)-C(x):
P(5.6)=(5.6)(1/4(44-5.6))-5.6^2+3(5.6)-64

2. Originally Posted by driver327
1. The company's cost C(x)=x^2-3x+64 dollars to produce x items The selling price p when x hundred units are produced is p(x)=1/4(44-x). Determine the level of production (# of items produced) that maximizes profit.
Here are my steps:

R(x)= x(1/4)(44-x)=11x-(1/4)x^2
R(x)=C(x)
11x-(1/4)^2=x^2-3x+64
derive: 11-(1/2)(x)=2x-3
-(1/2)x-2x=-14
-5/2(x)=-14
5x=28
x=28/5=5.6
Since Profit is P(x)= R(x)-C(x):
P(5.6)=(5.6)(1/4(44-5.6))-5.6^2+3(5.6)-64
The profit is P(x)=R(x)-C(x). You need to maximise the profit, so
you will be looking for the solutions of:

P'(x) = R'(x)-C'(x) = 0.

Then you will have to confirm that this solution is indeed a maximum.

RonL

3. You solution is correct. As the graph shows the is something wrong with the statement.

4. Originally Posted by driver327
1. The company's cost C(x)=x^2-3x+64 dollars to produce x items The selling price p when x hundred units are produced is p(x)=1/4(44-x). Determine the level of production (# of items produced) that maximizes profit.
Here are my steps:

R(x)= x(1/4)(44-x)=11x-(1/4)x^2
It says the the price is 1/4(44-x) for x hundred units. So shouldn't it be

R(x) = x(1/4)(44-x/100)=11x-(1/400)x^2?
R(x)=C(x)
This is not true. There is no reason R(x) = C(x). To maximize profits, set

P'(x) = R'(x) - C'(x) = 0.

This is what you end up with, but it does not come from R(x) = C(x).

11x-(1/4)^2=x^2-3x+64
derive: 11-(1/2)(x)=2x-3
-(1/2)x-2x=-14
-5/2(x)=-14
5x=28
x=28/5=5.6
Since Profit is P(x)= R(x)-C(x):
P(5.6)=(5.6)(1/4(44-5.6))-5.6^2+3(5.6)-64
What is the right answer? I get this.

Setting P'(x) = 11 - 2/400x - 2x + 3 = 0,
-(1/200+2)x = -14,
x = 2800/401 = 6.98.
P(x) = 11x-(1/400)x^2 - x^2+3x- 64 = -15.12.

5. JakeD
I think that the poster simply had a typo.
It is true that maximum profit occurs if marginal revenue equals marginal cost: R’=C’ & R”<C”.
I think that is what was meant, even if it was misrepresented.

I also think that there are errors in your calculations. I get the same answers a were posted.

6. ok, I used the P'=R'-C'=0.
(44x-x^2/4)-(x^2-3x+64)
derive R'= (-(4*(44-2x))/4^2)-(2x-3)=0 Because the 4 becomes a zero.
take out 2 from num and denom.
(-(2*(22-x)/8)-(2x-3)=0
(-(2*(22-x)/8)=2x-3
-(2*(22-x)=16x-24
-44+2x=16x-24
-44=14x-24
-20=14x
-10/7=x
-1.4285...=x
What else did I do wrong?

Also, I double read it for typos, That is what it said word for word.

7. Originally Posted by driver327
1. The company's cost C(x)=x^2-3x+64 dollars to produce x items The selling price p when x hundred units are produced is p(x)=1/4(44-x).
Originally Posted by JakeD
It says the the price is 1/4(44-x) for x hundred units. So shouldn't it be

R(x) = x(1/4)(44-x/100)=11x-(1/400)x^2?

What is the right answer? I get this.

Setting P'(x) = 11 - 2/400x - 2x + 3 = 0,
-(1/200+2)x = -14,
x = 2800/401 = 6.98.
P(x) = 11x-(1/400)x^2 - x^2+3x- 64 = -15.12.
Originally Posted by Plato
JakeD
I think that the poster simply had a typo.
It is true that maximum profit occurs if marginal revenue equals marginal cost: R’=C’ & R”<C”.
I think that is what was meant, even if it was misrepresented.

I also think that there are errors in your calculations. I get the same answers a were posted.
Plato,

The problem statement says cost C(x) is for x items while price p(x) is for x hundred units. So the price for x units would be

p(x/100) = (1/4)(44-x/100)

and revenue would be

R(x) = xp(x/100) = x(1/4)(44-x/100)=11x-(1/400)x^2.

I solved the problem under this assumption, so I got a different answer than you and driver327.

8. I found the value for x. 68/18=x=3.77778. mult. by a 100, i got 378 roughly. I plugged 378 into P(378)=R(378)-C(378)=-173377.
R((378)=(378(1/4)(44-378)-C=(378)^2-3(378)+64 then I got the answer -173377. If this is not right, I officially give up.