What is the marginal weekly cost at production level 142 sleeping bags per week? What is the average weekly cost per bag when the demand is 143? Yaster Outfitters manufactures and sells extreme-cold sleeping bags. The table below shows theprice-demand and total cost data, where:• pis the wholesale price in dollars) of a sleeping bag for a weekly demand of r sleeping bags:. Cis the total cost (in dollars) of producing z sleeping bags.z (sleeping bags)P (S)C (S)9524013,00012023514,30018015518,5002205021.000Price ModelFind a quadratic regression equation for the price-demand data, using z as the independent variable.p-at br + czRound a to the nearest integer, round b to 2 decimal places, and round e to 4 decimal places.Revenue ModelUsing the regression model computed above, find a model for the weekly revenue, using a as theindependent variable.NOTE: Do not calculate another regression. Use the price equation to find a model for revenueR(2) =p.z.R(2) - p.z (a + br + cz')z az + bz + czCost ModelFind a linear regression model for the weekly cost data, using a as the independent variable.C(2) mz +kRound m to 1 decimal place, and round k to the nearest integer.What is the marginal weekly cost at production level 142 sleeping bags per week? Round to 1decimal place.$.per sleeping bagUse the cost model to estimate the average weekly cost per bag when the weekly demand is 143.Round to the nearest cent.24per sleeping bagvestigate

Given: x (sleeping bags) p ($) C ($) 95 240 13000 120 235 14300 180 155 18500 220 50 21000 Here, p is the wholesale price (in dollars) of a sleeping bag for a weekly demand of x sleeping bags C is the total cost (in dollars) of producing x sleeping bags To determine: The marginal weekly cost at production level 142 sleeping bags per week (rounded to 1 decimal place) The average weekly cost per bag when the weekly demand is 143 (rounded to the nearest cent)Let the quadratic regression equation for the given price demand data, using x as the independent variable be, p=a+bx+cx2 Then, pi=a+bxi+cxi2 for some wholesale price pi for a weekly demand of xi sleeping bags We know that the matrix equation for the coefficients of the above quadratic regression equation is given as, ∑xi4∑xi3∑xi2∑xi3∑xi2∑xi∑xi2∑xincba=∑xi2pi∑xipi∑pi Here, n denotes the number of data points. Now, from the given table, xi pi xi2 xi3 xi4 xipi xi2pi 95 240 9025 857375 81450625 22800 2166000 120 235 14400 1728000 207360000 28200 3384000 180 155 32400 5832000 1049760000 27900 5022000 220 50 48400 10648000 2342560000 11000 2420000 ∑xi=615 ∑pi=680 ∑xi2=104225 ∑xi3=19065375 ∑xi4=3681130625 ∑xipi=89900 ∑xi2pi=1,29,92,000 Here, n=4. Then, the three equations obtained from these values and the matrix equation are: 3681130625c+19065375b+104225a=12992000 19065375c+104225b+615a=89900 104225c+615b+4a=680 368113062519065375104225190653751042256151042256154cba=1299200089900680 Then, cba=368113062519065375104225190653751042256151042256154-11299200089900680 Calculating inverse, we have, cba=0.0000001098292570373788636-0.0000345408398707891089930.0024489155514536224929-0.0000345408398707891089920.010966374995562812702-0.786075396684533731880.0024489155514536224925-0.7860753966845337317657.2995364026836101921299200089900680=1.4269017074296261958912-3.1052215043839408984707+1.665262574988463295172-448.754591601292104024064+985.8771121010968619098-534.531269745482937678431816.31084448546342256-70668.178161939582485224+38,963.68475382485493056=-0.01305722196585140740752.591250754321820207336111.817436380735867896 So, cba=-0.01305722196585140740752.591250754321820207336111.817436380735867896 This implies that, c=-0.0130572219658514074075, b=2.591250754321820207336 and a=111.817436380735867896 Rounding a to nearest integer, b to 2 decimal places and c to 4 decimal places, we have, c=-0.0131, b=2.59 and a=112 Put these values in p=a+bx+cx2 to get the regression equation as, p=112+2.59x-0.0131x2Now, the wholesale price (in dollars) of a sleeping bag for a weekly demand of x sleeping bags is modelled by the obtained regression equation, p=112+2.59x-0.0131x2. The weekly revenue function is then given as, Rx=p·x Put p=112+2.59x-0.0131x2 in Rx=p·x to get, Rx=112+2.59x-0.0131x2x=112x+2.59x2-0.0131x3 So, the weekly revenue function is Rx=112x+2.59x2-0.0131x3.Let the linear regression model for the weekly cost data, using x as the independent variable be, Cx=mx+k Then, Ci=mxi+k for some total cost Ci for producing xi sleeping bags in a week We know that the formulae for the coefficients of the above regression equation are given by: m=n∑xiCi-∑xi∑Cin∑xi2-∑xi2 k=∑Ci∑xi2-∑xi∑xiCi∑xi2-∑xi2 Here, n denotes the number of data points. Now, from the given table, xi Ci xi2 xiCi 95 13000 9025 1235000 120 14300 14400 1716000 180 18500 32400 3330000 220 21000 48400 4620000 ∑xi=615 ∑Ci=66800 ∑xi2=104225 ∑xiCi=10901000 Here, n=4. Using these values in the above formulae, the coefficients are evaluated as: m=410901000-615668004104225-6152=43604000-41082000416900-378225=252200038675=65.210084 k=66800104225-61510901000104225-6152=6962230000-6704115000104225-378225=258115000-274000≈-942.02554744525 Rounding m to 1 decimal place and k to the nearest integer, we have, m=65.2 and k=-942 Put these values in Cx=mx+k to get the regression equation as, Cx=65.2x-942Now, as obtained previously, the weekly cost function is given as Cx=65.2x-942 The marginal cost function is the derivative of the cost function. So, the marginal weekly cost function is evaluated as, Mx=ddxCx=ddx65.2x-942=65.2 So, the marginal weekly cost is a constant value of 65.2. Thus, the marginal weekly cost at production level 142 sleeping bags per week (rounded to 1 decimal place) is $65.2 per sleeping bag.Now, if the cost function is given as Cx, then the average cost is evaluated as Cx=Cxx. Here, we have determined that Cx=65.2x-942 Put Cx=65.2x-942 in Cx=Cxx to get the average cost function as, Cx=65.2x-942x=65.2-942x So, Cx=65.2-942x Put x=143 in Cx=65.2-942x to get, C143=65.2-942143≈65.2-6.5874125874125≈58.6125874125 Rounding off to two decimal places, we have, C143=58.61 This implies that the average weekly cost per bag when the weekly demand is 143 (rounded to the nearest cent) is $58.61 per sleeping bag.Answer: The marginal weekly cost at production level 142 sleeping bags per week (rounded to 1 decimal place) is $65.2 per sleeping bag. The average weekly cost per bag when the weekly demand is 143 (rounded to the nearest cent) is $58.61 per sleeping bag.

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What is the marginal weekly cost at production level 142 sleeping bags per week? What is the average weekly cost per bag when the demand is 143?

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Question

What is the marginal weekly cost at production level 142 sleeping bags per week? What is the average weekly cost per bag when the demand is 143?

Yaster Outfitters manufactures and sells extreme-cold sleeping bags. The table below shows the
price-demand and total cost data, where:
• pis the wholesale price in dollars) of a sleeping bag for a weekly demand of r sleeping bags:
. Cis the total cost (in dollars) of producing z sleeping bags.
z (sleeping bags)
P (S)
C (S)
95
240
13,000
120
235
14,300
180
155
18,500
220
50
21.000
Price Model
Find a quadratic regression equation for the price-demand data, using z as the independent variable.
p-at br + cz
Round a to the nearest integer, round b to 2 decimal places, and round e to 4 decimal places.
Revenue Model
Using the regression model computed above, find a model for the weekly revenue, using a as the
independent variable.
NOTE: Do not calculate another regression. Use the price equation to find a model for revenue
R(2) =p.z.
R(2) - p.z (a + br + cz')z az + bz + cz
Cost Model
Find a linear regression model for the weekly cost data, using a as the independent variable.
C(2) mz +k
Round m to 1 decimal place, and round k to the nearest integer.
What is the marginal weekly cost at production level 142 sleeping bags per week? Round to 1
decimal place.
$.
per sleeping bag
Use the cost model to estimate the average weekly cost per bag when the weekly demand is 143.
Round to the nearest cent.
24
per sleeping bag
vestigate

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