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| | & = & 116x-3x^{2}-(x^{2}+20x+64)\\ | | & = & 116x-3x^{2}-(x^{2}+20x+64)\\ |
| | \\ | | \\ |
| − | & = & -4x^{2}+136x+64. | + | & = & -4x^{2}+96x+64. |
| | \end{array}</math> | | \end{array}</math> |
| | | | |
| | To find the maximum value, we need to find a root of the derivative: | | To find the maximum value, we need to find a root of the derivative: |
| | | | |
| − | ::<math>0\,=\,P'(x)\,=\,-8x+136\,=\,-8(x-17),</math> | + | ::<math>0\,=\,P'(x)\,=\,-8x+96\,=\,-8(x-12),</math> |
| | | | |
| − | which has a root at <math style="vertical-align: -1px">x=17</math>. Plugging this into our function for profit, | + | which has a root at <math style="vertical-align: -1px">x=12</math>. Plugging this into our function for profit, |
| | we have | | we have |
| | | | |
| − | ::<math>P(17)\,=\,-4(17)^{2}+136(17)+64\,=\,1220.</math> | + | ::<math>P(12)\,=\,-4(12)^{2}+96(12)+64\,=\,640.</math> |
| | |} | | |} |
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| | |- | | |- |
| | | | | | |
| − | :'''(c)''' The maximum profit of <math style="vertical-align: -1px">1220</math> occurs at a production level of <math style="vertical-align: -1px">17</math> units. | + | :'''(c)''' The maximum profit of <math style="vertical-align: -1px">640</math> occurs at a production level of <math style="vertical-align: -1px">12</math> units. |
| | |- | | |- |
| | |Note that monetary units were not provided in the statement of the problem. | | |Note that monetary units were not provided in the statement of the problem. |
Revision as of 13:49, 5 December 2016
Given demand Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = 116 - 3x}
, and cost Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = x^2 + 20x + 64}
, find:
- a) Marginal revenue when x = 7 units.
- b) The quantity (x-value) that produces minimum average cost.
- c) Maximum profit (find both the x-value and the profit itself).
| Foundations:
|
| Recall that the demand function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)}
, relates the price per unit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}
to the number of units sold, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
.
Moreover, we have several important important functions:
|
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(x)}
, the total cost to produce Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
units;
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x)}
, the total revenue (or gross receipts) from producing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
units;
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)}
, the total profit from producing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
units;
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{C}(x)}
, the average cost of producing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
units.
|
| In particular, we have the relations
|
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)\,=\,R(x)-C(x),}
|
| while
|
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x)\,=\,x\cdot p(x),}
|
| and
|
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{C}(x)\,=\,\frac{C(x)}{x}.}
|
The marginal profit at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0}
units is defined to be the effective profit of the next unit produced, and is precisely  . Similarly, the marginal revenue or marginal cost would be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R'(x_0)}
or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C'(x_0)}
, respectively.
On the other hand, any time they speak of minimizing or maximizing, we need to find a local extrema. These occur when the first derivative is zero.
|
Solution:
| (a):
|
The revenue function is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x)\,=\, x\cdot p(x)\,=\, x(116-3x)\,=\,116x-3x^{2}}
.
Thus, the marginal revenue at a production level of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7}
units is simply
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R'(7)\,=\,116-6x\bigg|_{x=7}\,=\,116-6(7)\,=\,74.}
|
| (b):
|
We have that the average cost function is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \overline{C}(x) & = & {\displaystyle {\displaystyle \frac{C(x)}{x}}}\\ \\ & = & {\displaystyle {\displaystyle \frac{x^{2}+20x+64}{x}}}\\ \\ & = & {\displaystyle x+20+\frac{64}{x}.} \end{array}}
Our first derivative is then
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{C}\,'(x)\,=\,1-\frac{64}{x^{2}}\,=\,\frac{x^{2}-64}{x^{2}}\,=\,\frac{(x-8)(x+8)}{x^{2}}.}
This has a single positive root at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=8}
, which will correspond to the minimum average cost.
|
| (c):
|
First, we find the equation for profit. Using part of (a), we have
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} P(x) & = & {\displaystyle {\displaystyle R(x)-C(x)}}\\ \\ & = & 116x-3x^{2}-(x^{2}+20x+64)\\ \\ & = & -4x^{2}+96x+64. \end{array}}
To find the maximum value, we need to find a root of the derivative:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\,=\,P'(x)\,=\,-8x+96\,=\,-8(x-12),}
which has a root at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=12}
. Plugging this into our function for profit,
we have
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(12)\,=\,-4(12)^{2}+96(12)+64\,=\,640.}
|
| Final Answer:
|
- (a) The marginal revenue at a production level of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7}
units is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 74}
.
|
- (b) The minimum average cost occurs at a production level of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8}
units.
|
- (c) The maximum profit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 640}
occurs at a production level of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12}
units.
|
| Note that monetary units were not provided in the statement of the problem.
|
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