Difference between revisions of "022 Exam 2 Sample B, Problem 8"

Latest revision as of 16:32, 17 May 2015

Find the quantity that produces maximum profit, given demand function $p=70-3x$ and cost function $C=120-30x+2x^{2}.$

Foundations:
Recall that the demand function, $p(x)$ , relates the price per unit $p$ to the number of units sold, $x$ . Moreover, we have several important important functions:
$C(x)$ , the total cost to produce $x$ units; $R(x)$ , the total revenue (or gross receipts) from producing $x$ units; $P(x)$ , the total profit from producing $x$ units.
In particular, we have the relations
$P(x)=R(x)-C(x),$
and
$R(x)=x\cdot p(x).$
Using these equations, we can find the maximizing production level by determining when the first derivative of profit is zero.

Solution:

Step 1:
Find the Profit Function: We have
$R(x)\,=\,x\cdot p(x)\,=\,x\cdot (70-3x)\,=\,70x-3x^{2}.$
From this,
$P(x)\,=\,R(x)-C(x)\,=\,70x-3x^{2}-\left(120-30x+2x^{2}\right)\,=\,-120+100x-5x^{2}.$

Step 2:
Find the Maximum: The equation for profit at a given production level is
$P(x)\,=\,-120+100x-5x^{2}.$
Applying our power rule to each term, we find
$P'(x)\,=\,100-10x\,=\,10(10-x).$
The only root of this occurs at $x=10$ , and this is our production level to achieve maximum profit.

Final Answer:
Maximum profit occurs when we produce 10 items.

@@ Line 34: / Line 34: @@
 |-
 |
-::<math>R(x)\,=\,x\cdot p(x)\,=\,x\cdot (90-3x)\,=\,90x-3x^2.</math>
+::<math>R(x)\,=\,x\cdot p(x)\,=\,x\cdot (70-3x)\,=\,70x-3x^2.</math>
 |-
 |From this,
 |-
 |
-::<math>P(x)\,=\,R(x)-C(x)\,=\,90x-3x^2- \left(200-30x+x^2 \right)\,=\,120x-4x^2-200 .</math>
+::<math>P(x)\,=\,R(x)-C(x)\,=\,70x-3x^2- \left(120 - 30x + 2x^2 \right)\,=\,-120 + 100 x - 5 x^2 .</math>
 |}
@@ Line 45: / Line 45: @@
 !Step 2: &nbsp;
 |-
-|'''Find the Maximum:''' The equation for marginal revenue is
+|'''Find the Maximum:''' The equation for profit at a given production level is
 |-
 |
-::<math>P(x)\,=\,120x-4x^2-200 .</math>
+::<math>P(x)\,=\,-120 + 100 x - 5 x^2 .</math>
 |-
 |Applying our power rule to each term, we find
 |-
 |
-::<math>P'(x)\,=\,120-8x\,=\,8(15-x).</math>
+::<math>P'(x)\,=\,100-10x\,=\,10(10-x).</math>
 |-
-|The only root of this occurs at <math style="vertical-align: -5%">x=15</math>, and this is our production level to achieve maximum profit.
+|The only root of this occurs at <math style="vertical-align: -3%">x=10</math>, and this is our production level to achieve maximum profit.
 |-
 |
@@ Line 64: / Line 64: @@
 !Final Answer: &nbsp;
 |-
-|Maximum profit occurs when we produce 15 items.
+|Maximum profit occurs when we produce 10 items.
 |}
 [[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']]