5

(Average cost and marginal cost) If it costs a manufacturer $C(x)$ dollars to produce $x$ items, then his average cost of production is $C(x) / x$ dollars per item....

Question

(Average cost and marginal cost) If it costs a manufacturer $C(x)$ dollars to produce $x$ items, then his average cost of production is $C(x) / x$ dollars per item. Typically the average cost is a decreasing function of $x$ for small $x$ and an increasing function of $x$ for large $x$. (Why?) Show that the value of $x$ that minimizes the average cost makes the average cost equal to the marginal cost.

(Average cost and marginal cost) If it costs a manufacturer $C(x)$ dollars to produce $x$ items, then his average cost of production is $C(x) / x$ dollars per item. Typically the average cost is a decreasing function of $x$ for small $x$ and an increasing function of $x$ for large $x$. (Why?) Show that the value of $x$ that minimizes the average cost makes the average cost equal to the marginal cost.



Answers

(a) If $ C(x) $ is the cost of producing $ x $ units of a commodity, then the average cost per unit is $ c(x) = C(x)/x $. Show that if the average cost is a minimum, then the marginal cost equals the average cost.
(b) If $ C(x) = 16,000 + 200x + 4x^{3/2} $, in dollars, find (i) the cost, average costs, and marginal costs at a production level of $ 1000 $ units; (ii) the production leve that will minimize the average cost; and (iii) the minimum average cost.

This question asked us to show that the marginal cost equals the average cost and asked us to find the cost. Average cost and marginal cost is those production level and minimum average cost. What we know is that at the minimum we have C prime of X equals zero. Therefore, we know we have acts, see prime of acts minus sea of X equals zero. Therefore, what this means is the axe times or C of acts is equivalent to see of acts divided by ex not other words. Given the fact that seat prime of access equivalent to zero no see prime of X can be set equal to see of acts. Remember, we're looking for the marginal cost given the idea of sea of acts divided by acts, as you can see, No, we know, given the cost function, the capital C of acts no can pop. We can plug in C one thousands. We complacency of 1000. We have 342491 We're plugging this into capital C of acts. Therefore we know sea of 1000 divided by 1000 is this value is the value of three sea of 1000 which is 342491 divide by 1000. Remember, this is average cost, which is total cost divide by the number of units. This is 349 plus 491 So this is average cost right over here in level is average cost. Now we know marginal cost is gonna be see prime of 1000 which is gonna be 200 plus six times 2 1001 half, which is 3 89.74 This is marginal cost. Now we need to find whether where the marginal cost will equal the average cost. This is where see, prime of acts is cfx over X. We wrote this in the very first slide for part a consolidating or like terms we're get X is 400 now remember that the minimum average cost is going to be the critical value. We learn critical values in the context of our derivatives. When we were essentially solving for what X equals so plugging in we have 16,000 divided by 400 which is added to 200 and then plus four times 400 to the 1/2 were simply plugging us into the equation. CIA 400. Divide by 400 we end up with 320. This is the minimum average cost.

So the average course function has given my see eggs because to see X Over X and the mark, you know, cost function is given by its spirited. When she using the court room, it's he could do. Don't have C Times X miners dear to eggs times see over X square. So this is just X time. See your favorite books minus D eggs over X square.

Okay. So just so you know, the average cost per unit like this fancied see bar and thanks is going to be receive X function divided by X. We're over. So the average cost per unit, it's going to be the function that you were given all over x. No, no. Find that minimum. We need to find the derivative of dysfunction we just created, So we will probably have to use the quotient rule to help us to find that to repeated. So the derivative of our average value going to be represented by the following. So here is my derivative. Now, to find critical points and help us find minimums, Max, moments, whatever. We need to set this equal to zero in the soul for critical points. So just noticed that we just worry about the numerator. When will the numerator equal zero in Seoul? Another critical value that we don't have to worry about us much. Zero, since we cannot divide by zero without function of an existent or original anyway, So we don't have to worry about zero. So now 80 is a critical values. So here's 80 on the number line. Let's test something to elect a bit like the road and see what kind of number we get. My plug. Zero in. I'm gonna get a negative number. I played 100 in. We'll get a positive number. So that means that your average price is going to decrease until you get to 80 and then it will increase after 80. So the production number of X equals 80. And since it's going from negative thio or decreasing to increasing, this is a minimum average cost treatment.

Here. We're told that a sea of X is a cost of producing X units, then see of X over axes, the average cost per unit for part A. We want to show that if the average costs is minimum, then the marginal cost is equal to the average cost. So here we just have to set the derivative of the average cost equal to zero and then solve. So that is the average cost. So when we go to differentiate, we can just use quotient room, get that that IHS X time see prime of X minus sea of X all over X squared and then setting that equal to zero we get that X time. See, Prime of Axe is equal to see a vax. And that means that see, Prime of X is equal to see of ex all over axe. And that's what That's just what we wanted because, see, prime of axe, that's our marginal cost and then see of X over X. That's the average cost. So that's our solution for party. And then when we get to Part B, were given the cost function 16,000 plus 200 acts plus four times X to the three halfs and we want to find for part one we want to find the cost, the average cost and the marginal cost at 1000 units. We know that the cost was given to us and that's just see of 1000 and that is approximately 342,000 491. And so that's our cost. And then we want the average cost. So that's just see of 1000 all over 1000 and thats approximately 342 point for nine. And then next we want the marginal cost and the marginal cost is just see prime of 1000. So when we differentiate our cost function, we get 200 plus three halfs times, for which is six times the square root of 1000 and thats approximately equal to 389 0.74 And that was our marginal cost at 1000 units. So that is the solution for B. We'll call this be one and now the second part, ask us. It's asking us to find the production level that minimizes theatrics cost, So here we just do the same thing. We differentiate the average cost. But to do that, let's first start with finding the average costs. We know that that's just Segovax over axe and see if X was given to us. So that's 16,000 plus 200 acts plus four times X to the three halfs all over acts. And then we can break this up. We get that that 16,000 over X plus 200 plus four times X to the 1/2. And now when we take the derivative of our average costs, we get that that's minus 16,000 over X squared, plus two all over the square root of X. We want to set that equal to zero. So now we have that 16,000 all over. A tune is equal to X square and all over route acts, which means that 8000 is equal to X to the three halfs and then solving for X, we get that X is equal to 400. And that's the solution for part two of being, because we just wanted to find the production level that minimizes that average cost, and we know that that is a minimum for the following reason. So if we take the limit as X approaches zero of this function, we get that that's infinite, that's infinity, and then the limit as X approaches. Infinity of this function is also infinity. So that means that the value, the critical value that we found must be a minimum value. And so now the last part says to find the minimum average cost. So here we just have toe take CF 400 all over 400 because X equals 400 is where the minimum occurs. And if we do that, we get that that is equal to 320 and that completes the problem.


Similar Solved Questions

5 answers
For the decomposition of dinitrogen tetroxide: NO(g) = 2NO(g) Write down the expresion of KeKc0.31b) Ifa reaction mixture at equilibrium contains [N,Oa]-0.4SM and knowing that Kc-0.31, calculate [NO]
For the decomposition of dinitrogen tetroxide: NO(g) = 2NO(g) Write down the expresion of Ke Kc0.31 b) Ifa reaction mixture at equilibrium contains [N,Oa]-0.4SM and knowing that Kc-0.31, calculate [NO]...
4 answers
Find 013 and Czz, vhere2A 38,333] and B = 3 -1|' -5Need Help?Read ItIkte_ Tutor[Show My Work (Requlred) What steps Or reasoning did Yoj Lie? Your work Coln? towaro; your 3core You can gubmit ahow my "cr unlimite numbe ~fiimes
Find 013 and Czz, vhere 2A 38, 333] and B = 3 -1|' -5 Need Help? Read It Ikte_ Tutor[ Show My Work (Requlred) What steps Or reasoning did Yoj Lie? Your work Coln? towaro; your 3core You can gubmit ahow my "cr unlimite numbe ~fiimes...
5 answers
Soiving appucO Piobictshalf-life tte body of L5 Suppase jadent tazes one 200. mg plll at 5 (uph 210 another identical pill 45 min later: Calculate tte cettaln dnyg has Inount dna left his tcdy at 7 COPM.sure Ycur anser hussymbol nexessary; and rourd $ t0 sign fican: diqies;
Soiving appucO Piobicts half-life tte body of L5 Suppase jadent tazes one 200. mg plll at 5 (uph 210 another identical pill 45 min later: Calculate tte cettaln dnyg has Inount dna left his tcdy at 7 COPM. sure Ycur anser hus symbol nexessary; and rourd $ t0 sign fican: diqies;...
5 answers
Convert the binary expansion of each of these integers to a decimal expansion.a) $(11111)_{2}$b) $(1000000001)_{2}$c) $(101010101)_{2}$d) $(110100100010000)_{2}$
Convert the binary expansion of each of these integers to a decimal expansion. a) $(11111)_{2}$ b) $(1000000001)_{2}$ c) $(101010101)_{2}$ d) $(110100100010000)_{2}$...
5 answers
The sixth term and the eleventh term of an arithmetic progression are 30 and 55 respectively. Find the twentyfirst term of the series.(A) $88 rac{1}{3}$(B) 105(C) 110(D) $92 rac{1}{2}$
The sixth term and the eleventh term of an arithmetic progression are 30 and 55 respectively. Find the twentyfirst term of the series. (A) $88 \frac{1}{3}$ (B) 105 (C) 110 (D) $92 \frac{1}{2}$...
1 answers
Perform the addition or subtraction and use the fundamental identities to simplify. $$\frac{1}{\sec x+1}-\frac{1}{\sec x-1}$$
Perform the addition or subtraction and use the fundamental identities to simplify. $$\frac{1}{\sec x+1}-\frac{1}{\sec x-1}$$...
5 answers
Aeida plarie electromagnetic wave has maximum electuc ficld magnitude amplitude.10*3 Vlm Find the maximum magneticNumberUnitsthe tolerance I 4/-200
Aeida plarie electromagnetic wave has maximum electuc ficld magnitude amplitude. 10*3 Vlm Find the maximum magnetic Number Units the tolerance I 4/-200...
5 answers
Q1. Determine the directional derivative of v2 where vexyi+zy2j+*z? k at the point (2, 0, 3) in the direction of (he outward normal to the sphere x+y? + 22 =14 at (he point (3,2, 1).
Q1. Determine the directional derivative of v2 where vexyi+zy2j+*z? k at the point (2, 0, 3) in the direction of (he outward normal to the sphere x+y? + 22 =14 at (he point (3,2, 1)....
5 answers
Which is the derivative for yANz8?32dd d d8 552 Vzs32 c7d
Which is the derivative for y ANz8? 32 d d d d 8 55 2 Vzs 32 c7 d...
5 answers
ParACalculate the kinetic energy in J ol an electron moving al 6.00 L0" rs, The mass 0l an eleciton Is 9.11 > I0r6.56 * 10-141843.284.98249SubmnilRequezt Anwcr
ParA Calculate the kinetic energy in J ol an electron moving al 6.00 L0" rs, The mass 0l an eleciton Is 9.11 > I0r 6.56 * 10-14 184 3.28 4.98 249 Submnil Requezt Anwcr...
5 answers
3) simplitytront coontntSolyc lor x Expressyour final an5vct using Intenvol notettanI2x- 71>135] Considcr fx) grpbce brlovg. Usethc same coordinztc plane t0 erzph slr)= Ax* 3) + &Subtrzst and simplly: 7 %+2 07) Solve (orr;4
3) simplity tront coontnt Solyc lor x Expressyour final an5vct using Intenvol notettan I2x- 71>13 5] Considcr fx) grpbce brlovg. Usethc same coordinztc plane t0 erzph slr)= Ax* 3) + & Subtrzst and simplly: 7 %+2 0 7) Solve (orr; 4...
4 answers
Table 4. Pairwise comparison matrix developed for subbasin 43 based on Cost at the watershed outlet.BMPsSCRMCTNGNTSC RM CT NG NT1.92885906Step 2: All criteria have the same weightSR TN TP Cost0.25 0.25 0.251 0.251
Table 4. Pairwise comparison matrix developed for subbasin 43 based on Cost at the watershed outlet. BMPs SC RM CT NG NT SC RM CT NG NT 1.92885906 Step 2: All criteria have the same weight SR TN TP Cost 0.25 0.25 0.251 0.251...
5 answers
The equilibrium constant for the chemical equationNz(g) + 3Hz(g) ~ 2NH;(g)is Kp = 0.0641 at 217 *C. Calculate the value of Kc for the reaction at 217 'C.Kc
The equilibrium constant for the chemical equation Nz(g) + 3Hz(g) ~ 2NH;(g) is Kp = 0.0641 at 217 *C. Calculate the value of Kc for the reaction at 217 'C. Kc...
5 answers
A linear map T : Rt + R2 cant be one-to-one: (Hint: Think about the standard matrix of T) TrueFalse
A linear map T : Rt + R2 cant be one-to-one: (Hint: Think about the standard matrix of T) True False...
3 answers
Suppose that U1, 02 ,Ukare a spanning collection of vectors for RIlWhat must be true about the size ofkwith respect to n?Ok<n0k2n0 *=n
Suppose that U1, 02 , Ukare a spanning collection of vectors for RIl What must be true about the size ofkwith respect to n? Ok<n 0k2n 0 *=n...

-- 0.024465--