00010001010008main.ACT0001020013cost-rev-profit.EAC0100000053edÉø‹^tÓ×Ø[Cost, Revenue and [Profit Functions[\AuthoráखMa. Louise Antonette De Las Penas,Phd Associate Professor Ateneo de Manila UniversityQuezon City, Philippines e-mail:mlp@mathsci.math.admu.edu.ph ÿ[ Objective: [To study rates of change[involving economic [quantities such as cost,[revenue and profit [analytically,graphically [and geometrically. [[ Problem: [A manufacturing company[estimates that the cost[in dollars of producing [ í¸ items is [C(í¸)=18í¸2+3í¸+98,[where all units will be [sold if price is given by[p(í¸)=753-í¸3 dollars per[unit.[(a)How many units must[the manufacturer sell [to break even?[(b)Determine the [production level that [will maximize profit.[(c)Use marginal analysis[to estimate the profit [derived from the sale of[the 21st, 24th and 30th[units.[[ Solution:[(a) The revenue function[is computed as [R(í¸)=í¸îp(í¸)=[í¸î753-í¸3=75í¸3-í¸23[Algebraic Approach:[To solve for the break [even points algebraically,[we equate R(í¸)=C(í¸) [and solve for í¸ in the [calculation row as [follows:Rsolve(75í¸3-í¸23=18í¸2+3í¸+98)Rx= 4.968924934,x= 43.03107507[Graphical Approach:[By dragging the cost and[revenue functions to the["Graph Window" below, [we can determine the [point of intersection of [both curves, which are[the break-even-points.[Tap Analysis-G-Solve-[Intersect to obtain [í¸~4.97 units,í¸~43 units[[(b) In the calculation [row, we compute the [profit function P(í¸) as[follows:R simplify(75í¸3-í¸23-(18í¸2 +3í¸+98))R-11î’x224+22î’x-98[Graphical Approach:[[We drag the profit [function to the "Graph [Window" below to [calculate graphically the[maximum point.[Using Analysis-G-Solve-[Maximum, we obtain [ í¸=24 units.[\Graph Window îÌÊÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ«Ñ aseq_histbNewFolde systemä]listsystemä^]=system«Ñsystemä]listsystemä^]=system«Ñsystemä]listsystemä^]=system«Ñsystemä]listsystemä^]=system«Ñsystemä]listsystemä^]=system«Ñsystemä]listsystemä^]=system«Ñsystemä]listsystemä^]=system«Ñsystemä]listsystemä^]=system«Ñsystemä]listsystemä^]=system«Ñsystemä]listsystemä^]=system«Ñ  0@PSheet1”øÀšŒ^Sheet2”øÀšŒ^Sheet3”øÀšŒ^Sheet4”øÀšŒ^Sheet5”øÀšŒ^SheetSheet3D 0@PSheet1蚌^Sheet2蚌^Sheet3蚌^Sheet4蚌^Sheet5蚌^SheetSheet3D i™ ™ø‹^tÓ× $ `P`P`,ú,ú,ú$gS$gS ™,ú`,ú,ú,ú(1…0qy`,ú#Y‡uYƒ,ú ˜@B`q&1W‰O[[(c) Geometric Approach:[The marginal profit, [represented geometrically[by the tangent line to[the profit function P(í¸)[at í¸=a, is a good [approximation of P(a).[We generate the tangent[line to P(í¸) and [calculate its slope at [varying values of í¸ as[follows:[[In the Geometry Window,[we draw the profit [function and animate the[tangent line to the [curve.[Tap Geometry Window-[Edit-Animate-Go(once)[To generate the table of[slopes:select point on [curve,select table;then[select slope of tangent[line at point,then select[table.\Geometry WindowîÎΑy™™™C‘E C@i vTV`vrALCuxyÈÀ  1W‰Pcx”sh@%&1W1W‰G@7‰G6€D!&0PRc€V„!0cx”piG6„ ux”sp‚Rc ˆBR`”shB&1`6„!hBP ffx`w1`P‡„30’˜4P5†Pq“”™` ™’™ep ™…•(3 Y™ d&g`Y™T8Gfp`'—P`pB`(•3@`…–6ƒ `C…„fp`u4fp`YdP`T&`uCp™`337P`ÈÀ# 1W‰Pcx”sh@%&1W1W‰G@7‰G6€D!&0PRc€V„!0cx”piG6„ ux”sp‚Rc ˆBR`”shB&1`6„!hBP ffx`w1`P‡„30’˜4P5†Pq“”™` ™’™ep ™…•(3 Y™ d&g`Y™T8Gfp`'—P`pB`(•3@`…–6ƒ `C…„fp`u4fp`YdP`T&`uCp™`337P`ËÀ# 1W‰Pcx”sh@%&1W1W‰G@7‰G6€D!&0PRc€V„!0cx”piG6„ ux”sp‚Rc ˆBR`”shB&1`6„!hBP ffx`w1`P‡„30’˜4P5†Pq“”™` ™’™ep ™…•(3 Y™ d&g`Y™T8Gfp`'—P`pB`(•3@`…–6ƒ `C…„fp`u4fp`YdP`T&`uCp™`337P`ÊÀ! 1W‰Pcx”sh@%&1W1W‰G@7‰G6€D!&0PRc€V„!0cx”piG6„ ux”sp‚Rc ˆBR`”shB&1`6„!hBP ffx`w1`P‡„30’˜4P5†Pq“”™` ™’™ep ™…•(3 Y™ d&g`Y™T8Gfp`'—P`pB`(•3@`…–6ƒ `C…„fp`u4fp`YdP`T&`uCp™`337P` 5Æ@! ÆH!ƒ H@Ð6–°EeˆTff ÈÀ  22îx-11îx^2/24-98E 1W‰G6„ ™ .È@  3ÈÀ  ÈÀ  @Ð6–°dSQƒ ™ ‘' ™a‘!$F`.Ë@# .ÈH   1Ä@  9  [[From the animation, [observe that the tangent[line varies in slope as it[goes through the whole[curve.This shows where[profit is increasing and [ decreasing.[From the table below, [note that slope is [positive before í¸=24, [becomes approximately [zero at í¸=24, slope is[negative after.[Profit derived from the [sale of 21st, 24th and[30th units is respectively[3,0,-6. [[Table of xvalues & slope[20 3.666678166 20.63157895 3.087731166 21.26315789 2.508784333 21.89473684 1.929834501 22.52631579 1.350886501 23.15789474 0.7719394996 23.78947368 0.1929916657 24.42105263 -0.3859528332 25.05263158 -0.9649026676 25.68421053 -1.543847667 26.31578947 -2.1227975 26.94736842 -2.701742 27.57894737 -3.280695334 28.21052632 -3.859636832 28.84210526 -4.438584667 29.47368421 -5.017534667 30.10526316 -5.596481501 30.73684211 -6.175426 31.36842105 -6.75437099932 -7.333337501[[An alternative way to [study where profit is [increasing/decreasing is [graph the x values and[slopes obtained above in[the List Editor.[Drag the columns of x [values and slopes to [List Window and Tap [Graph. [Tap Analysis-Trace to go[from one point to the [next. Clearly, it can be[seen where profit is [increasing/decreasing[(above/below x-axis). [Profit derived from the [sale of 21st, 24th and[30th units is respectively[3,0,-6. 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Its roots are [approximately í¸~4.97,[í¸~43.[[2.Verify that at the [point where profit is [maximum,marginal revenue[=marginal cost[ Solution:[Drag the cost and [revenue functions to["Graph Window" to graph[both functions.[Tap Analysis-Sketch-[ Tangent. [Press 1,then enter the[x-value 24.Press EXE to[construct tangent line [to first curve;tap arrow[down then EXE to do [the same thing to the [ second curve.[Observe that the curves[have parallel tangents [at the production level[í¸=24 since marginal [revenue equals marginal[cost. The slope of both[tangent lines to the [curves,given by the [derivative is [approximately 9.[[Note: We can also check[that at a production [level of í¸=24, profit is[maximum since C''(24)>[R''(24) as we see below.[The C''(24) is shown[here:Rdiff(18í¸2 +3í¸+98,í¸,2)R14[The R''(24) is shown[here:Rdiff(75í¸3-í¸23,í¸,2)R-23[eAct