mathsci2.appstate.edumathsci2.appstate.edu/~osbornejm/MAT1030/QuizzesHomework...MAT$1030$$Exam$4$Version$1:$ $ Last$Name$_ $,$First$Name_ $ $ Circle$Your$SecAon$Number:$$$Sec.$112$(3F350)$$$or$Sec.$114$(4F450)$$$

MAT$1030$$Exam$4$Version$1:$$

Last$Name$________________________$,$First$Name__________________________$$Circle$Your$SecAon$Number:$$$Sec.$112$(3F350)$$$or$Sec.$114$(4F450)$$$

QuesAon$1:$Answer$the$following$quesAon$using$Excel$and/or$Maple.$$$$$$$$$$$$$$$$$$$$$!!Fill!in!your!answer!on!this!sheet!in!the!space!provided.!!!!!!!!!!!!!!!!!!!!!!!!Show!formulas!used,!where!appropriate,!to!receive!par5al!credit.!

Assume that the demand price for a certain gadget is $3.25/gadget and that you have collected the following average cost (per item) observations below.!!First, set up, in Excel, a spreadsheet with rows total cost, revenue, and profit. Now answer the following:!!(a)  Fit your profit data with a polynomial of order 2 and find your predicted profit in selling 190 gadgets. !(b)  What is your marginal profit for selling q=126 gadgets?!(c)  What is your marginal profit for selling q=190 gadgets?!(d)  Find the derivative of your profit function and use it to find P’(290).!(e)  Use your derivative to approximate the profit you will make by selling 291 gadgets. Hint: a=290 and h=1!(f)  How many gadgets do you need to sell to maximize your profit? What is your maximum profit?!!!!

!(a)  Profit Function=_____________________________________________________________! ! Predicted Profit in Selling 190 gadgets=___________________________________________ !!(b) Marginal Profit=_______________________________!!(c) Marginal Profit=________________________!!(d) P’(290)=________________________________!!(e) Approximate P(291)=_______________________________!!(f) # of gadgets to sell to maximize profit=____________________________________! ! Maximum Profit=________________________________________________!

DirecAons:$$Answer$the$quesAons$on$this$exam$using$Excel$and/or$Maple.$$$$$$$$$$$$$$$$$$$$$!!Choose!oNe!of!Ques9on!1!or!Ques9on!2!to!complete.!!

!!!!!!!!!!Everyone!must!complete!Ques9on!3.!!!!!!!!!!!!!!!!!!!!!!!Choose!oNe!of!Ques9on!4!or!Ques9on!5!to!complete.!

!!!!!!!!!!Choose!oNe!of!Ques9on!6!or!Ques9on!7!to!compete.!!!!!!!!!!!!!!!!!!!!!!!Show$formulas$used,$where$appropriate,$to$receive$parAal$credit.$

QuesAon$2:$Answer$the$following$quesAon$using$Excel$and/or$Maple.$$$$$$$$$$$$$$$$$$$$$!Fill!in!your!answer!on!this!sheet!in!the!space!provided.!!!!!!!!!!!!!!!!!!!!!!Show!formulas!used,!where!appropriate,!to!receive!par5al!credit.!!

A demand price function for a gadget X is given by p=-0.45q+120. Your company, which makes gadget X, !has determined that their costs for producing 180 gadgets is $175 and the cost for producing 210 gadgets is!$235. The company cannot make less that 100 gadgets and cannot make more than 300 gadgets.!!(a)  Find a profit function.!(b)  Find the quantity that maximizes profit. !(c)  Find the maximum profit.!(d)  What is your marginal profit in selling q=160 gadgets?!(e)  Using a derivative, approximate the profit you will make/lose by selling 181 gadgets. Hint: a=180 and h=1!

!(a) Profit Function=_______________________________________________!!(b) To Maximize Profit q=__________________________________________!!(c) Maximum Profit=______________________________________________!!(d) Marginal Profit=_______________________________________________!!(e) Approximate P(181)=___________________________________________ !!


Minimize E=3x^2+2y^2 subject to the constraint y^2=1-4x.!

!(a)  value which minimizes E, x=______________!

(b)  value which minimizes E, y=______________!

(c)  minimum value of E=_______________!


Suppose the two enclosures shown below are to be fenced using 5000 feet of fence. What values of w and l!maximize the total area of the two enclosures? What is the total maximum area enclosed?!

!(a)  value which maximizes total area, w=______________!

(b)  value which maximizes total area, l=______________!

(c)  maximum value of total area=_______________!


Suppose the two enclosures shown below are to enclose a total area of 62450 square feet. !What values of w and l lead to least amount of perimeter fencing? What is the total length of fence needed?!

!(a)  value which minimizes fence, w=______________!

(b)  value which minimized fence, l=______________!

(c)  total minimum length of fence=_______________!

QuesAon$5:!Answer!the!following!ques1ons!using!Excel!and/or!Maple.!!!!!!!!!!Fill!in!your!answer!on!this!sheet!in!the!space!provided.!

!!!!!!!!!!!!!!!!!!!!Show!formulas!used,!where!appropriate,!to!receive!par5al!credit.!!!!!You are building a barn and a fenced in animal enclosure according to the picture below. !

You have gone to Lowes or Home Depot to price out wood (for the barn) and fencing (for the enclosure)!and learned that barn building wood costs $26 per foot and that fencing costs $2 per foot. Before you !build the barn and enclosure you ask yourself what your restrictions are. First, you know that you!want to minimize the cost of building both the barn and the enclosure. Second, you know that you would like!to enclose an area of 1275 square feet. Also, note that the barn is 104 feet long. !What is the size of the enclosure that minimizes the cost? !What is the total minimum cost? !

(y)!

(x)!

2 ft. of barn !over-hang on !each side!

(a) Size of enclosure x by y=_______________________________!

(b) Total minimum cost of barn and enclosure=_______________________________!!

QuesAon$5:!Answer!the!following!ques1ons!using!Excel!and/or!Maple.!!!!!!!!!!Fill!in!your!answer!on!this!sheet!in!the!space!provided.!

!!!!!!!!!!!!!!!!!!!!!Show!formulas!used,!where!appropriate,!to!receive!par5al!credit.!!!!!

You want to build a house according to the picture below. If siding costs $5 per square foot and roofing! costs $14 per square foot then what dimensions h and l minimize the cost for the exterior of your house?!

(a)  h=_______________________________!

(b)  l=_______________________________!

(c) Total minimum cost of exterior=_______________________________!!

MAT$1030$$Exam$4$Version$2:$$

Last$Name$________________________$,$First$Name__________________________$$Circle$Your$SecBon$Number:$$$Sec.$112$(3F350)$$$or$Sec.$114$(4F450)$$$

QuesBon$1:$Answer$the$following$quesBon$using$Excel$and/or$Maple.$$$$$$$$$$$$$$$$$$$$$!!Fill!in!your!answer!on!this!sheet!in!the!space!provided.!!!!!!!!!!!!!!!!!!!!!!!!Show!formulas!used,!where!appropriate,!to!receive!par5al!credit.!

Assume that the demand price for a certain gadget is $4.25/gadget and that you have collected the following average cost (per item) observations below.!!First, set up, in Excel, a spreadsheet with rows total cost, revenue, and profit. Now answer the following:!!(a)  Fit your profit data with a polynomial of order 2 and find your predicted profit in selling 190 gadgets. !(b)  What is your marginal profit for selling q=126 gadgets?!(c)  What is your marginal profit for selling q=190 gadgets?!(d)  Find the derivative of your profit function and use it to find P’(290).!(e)  Use your derivative to approximate the profit you will make by selling 291 gadgets. Hint: a=290 and h=1!(f)  How many gadgets do you need to sell to maximize your profit? What is your maximum profit?!!!!

!(a)  Profit Function=_____________________________________________________________! ! Predicted Profit in Selling 190 gadgets=___________________________________________ !!(b) Marginal Profit=_______________________________!!(c) Marginal Profit=________________________!!(d) P’(290)=________________________________!!(e) Approximate P(291)=_______________________________!!(f) # of gadgets to sell to maximize profit=____________________________________! ! Maximum Profit=________________________________________________!

DirecBons:$$Answer$the$quesBons$on$this$exam$using$Excel$and/or$Maple.$$$$$$$$$$$$$$$$$$$$$!!Choose!oNe!of!Ques9on!1!or!Ques9on!2!to!complete.!!

!!!!!!!!!!Everyone!must!complete!Ques9on!3.!!!!!!!!!!!!!!!!!!!!!!!Choose!oNe!of!Ques9on!4!or!Ques9on!5!to!complete.!

!!!!!!!!!!Choose!oNe!of!Ques9on!6!or!Ques9on!7!to!compete.!!!!!!!!!!!!!!!!!!!!!!!Show$formulas$used,$where$appropriate,$to$receive$parBal$credit.$

QuesBon$2:$Answer$the$following$quesBon$using$Excel$and/or$Maple.$$$$$$$$$$$$$$$$$$$$$!Fill!in!your!answer!on!this!sheet!in!the!space!provided.!!!!!!!!!!!!!!!!!!!!!!Show!formulas!used,!where!appropriate,!to!receive!par5al!credit.!!

A demand price function for a gadget X is given by p=-0.35q+140. Your company, which makes gadget X, !has determined that their costs for producing 190 gadgets is $175 and the cost for producing 210 gadgets is!$235. The company cannot make less that 100 gadgets and cannot make more than 300 gadgets.!!(a)  Find a profit function.!(b)  Find the quantity that maximizes profit. !(c)  Find the maximum profit.!(d)  What is your marginal profit in selling q=160 gadgets?!(e)  Using a derivative, approximate the profit you will make/lose by selling 181 gadgets. Hint: a=180 and h=1!

!(a) Profit Function=_______________________________________________!!(b) To Maximize Profit q=__________________________________________!!(c) Maximum Profit=______________________________________________!!(d) Marginal Profit=_______________________________________________!!(e) Approximate P(181)=___________________________________________ !!


Minimize E=2x^2+3y^2 subject to the constraint y^2=1-5x.!

!(a)  value which minimizes E, x=______________!

(b)  value which minimizes E, y=______________!

(c)  minimum value of E=_______________!


Suppose the two enclosures shown below are to be fenced using 6000 feet of fence. What values of w and l!maximize the total area of the two enclosures? What is the total maximum area enclosed?!

!(a)  value which maximizes total area, w=______________!

(b)  value which maximizes total area, l=______________!

(c)  maximum value of total area=_______________!


Suppose the two enclosures shown below are to enclose a total area of 60450 square feet. !What values of w and l lead to least amount of perimeter fencing? What is the total length of fence needed?!

!(a)  value which minimizes fence, w=______________!

(b)  value which minimized fence, l=______________!

(c)  total minimum length of fence=_______________!

QuesBon$5:!Answer!the!following!ques1ons!using!Excel!and/or!Maple.!!!!!!!!!!Fill!in!your!answer!on!this!sheet!in!the!space!provided.!

!!!!!!!!!!!!!!!!!!!!Show!formulas!used,!where!appropriate,!to!receive!par5al!credit.!!!!!You are building a barn and a fenced in animal enclosure according to the picture below. !

You have gone to Lowes or Home Depot to price out wood (for the barn) and fencing (for the enclosure)!and learned that barn building wood costs $20 per foot and that fencing costs $3 per foot. Before you !build the barn and enclosure you ask yourself what your restrictions are. First, you know that you!want to minimize the cost of building both the barn and the enclosure. Second, you know that you would like!to enclose an area of 1075 square feet. Also, note that the barn is 104 feet long. !What is the size of the enclosure that minimizes the cost? !What is the total minimum cost? !

(y)!

(x)!

2 ft. of barn !over-hang on !each side!

(a) Size of enclosure x by y=_______________________________!

(b) Total minimum cost of barn and enclosure=_______________________________!!

QuesBon$5:!Answer!the!following!ques1ons!using!Excel!and/or!Maple.!!!!!!!!!!Fill!in!your!answer!on!this!sheet!in!the!space!provided.!

!!!!!!!!!!!!!!!!!!!!!Show!formulas!used,!where!appropriate,!to!receive!par5al!credit.!!!!!

You want to build a house according to the picture below. If siding costs $7 per square foot and roofing! costs $16 per square foot then what dimensions h and l minimize the cost for the exterior of your house?!

(a)  h=_______________________________!

(b)  l=_______________________________!

(c) Total minimum cost of exterior=_______________________________!!

Scrap Paper If You Need!




> >

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Question 1, Version 1

P q dK0.0002$q2 C 1.8377$qK 5.5294; MP 126 = P 127 KP 126 ; MP 190 = P 191 KP 190 ; " P' " = P ' q ; " P'(290) " = P ' 290 ; " P(291) approx " = P 290 CP ' 290 $1; " P(291) exact " = P 291 ; plot P q , q = 0 ..8000 : Maxq = solve P ' q = 0, q , MaxProfit = P 4594.25

P := q/ K1 $0.0002 q2 C 1.8377 qK 5.5294MP 126 = 1.7871MP 190 = 1.7615

" P' " = K0.0004 qC 1.8377" P'(290) " = 1.7217

" P(291) approx " = 512.3053" P(291) exact " = 512.3051

Maxq = 4594.250000, MaxProfit = 4215.897213

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Question 1, Version 2

P q dK0.0002$q2 C 2.8377$qK 5.5294; MP 126 = P 127 KP 126 ; MP 190 = P 191 KP 190 ; " P' " = P ' q ; " P'(290) " = P ' 290 ; " P(291) approx " = P 290 CP ' 290 $1; " P(291) exact " = P 291 ; plot P q , q = 0 ..11000 :Maxq = solve P ' q = 0, q , MaxProfit = P 7094.25

P := q/ K1 $0.0002 q2 C 2.8377 qK 5.5294MP 126 = 2.7871MP 190 = 2.7615

" P' " = K0.0004 qC 2.8377" P'(290) " = 2.7217

" P(291) approx " = 803.3053" P(291) exact " = 803.3051

Maxq = 7094.250000, MaxProfit = 10060.14721

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Question2Version1; R q d expand q$ K0.45$qC 120 :

C q d 175C175K 235180K 210

$ qK 180 :

Revenue = R q ; Cost = C q ; P q d R q KC q : Profit = P q ; plot P q , q = 100 ..300 : Maxq = solve P ' q = 0, q , MaxProfit = P 131.11 ; MP 160 = P 161 KP 160 ; " P(181) approx " = P 180 CP ' 180 $1; " P(181) exact" = P 181 ;

Question2Version1

Revenue = K0.45 q2 C 120 qCost = K185C 2 q

Profit = K0.45 q2 C 118 qC 185Maxq = 131.1111111, MaxProfit = 7920.555555

MP 160 = K26.45" P(181) approx " = 6801.00" P(181) exact" = 6800.55

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Question2Version2; R q d expand q$ K0.35$qC 140 :

C q d 175C175K 235190K 210

$ qK 190 :

Revenue = R q ; Cost = C q ; P q d R q KC q : Profit = P q ; plot P q , q = 100 ..300 :Maxq = solve P ' q = 0, q , MaxProfit = P 195.71 ;

MP 160 = P 161 KP 160 ; " P(181) approx " = P 180 CP ' 180 $1; " P(181) exact" = P 181 ;

Question2Version2

Revenue = K0.35 q2 C 140 qCost = K395C 3 q

Profit = K0.35 q2 C 137 qC 395Maxq = 195.7142857, MaxProfit = 13801.42856

MP 160 = 24.65" P(181) approx " = 13726.00" P(181) exact" = 13725.65

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> > Question3Version1; E x d 3$x2 C 2$ 1K 4$x ; plot E x , x =K1 ..2 :

Minx = solve E ' x = 0, x , Miny = sqrt 1K4$43

, MinE = E43

;

Minx = solve E ' x = 0., x , Miny = sqrt 1K4$43.

, MinE = E43.

;

Question3Version1

E := x/3 x2 C 2K 8 x

Minx =43

, Miny =13

I 39 , MinE = K103

Minx = 1.333333333, Miny = 2.081665999 I, MinE = K3.333333329

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Question3Version1; E x d 2$x2 C 3$ 1K 5$x ; plot E x , x =K1 ..5 :

Minx = solve E ' x = 0, x , Miny = sqrt 1K5$15

4, MaxE = E

154

;

Minx = solve E ' x = 0., x , Miny = sqrt 1K5$15

4., MaxE = E

154.

;

Question3Version1

E := x/2 x2 C 3K 15 x

Minx =154

, Miny =12

I 71 , MaxE = K2018

Minx = 3.750000000, Miny = 4.213074887 I, MaxE = K25.12500000

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> > Question4Version1; Constraintd 3$wC 6$l = 5000;

A w d3$w$ 5000K 3$w

6:

Objective = A w ; plot A w , w = 0 ..900 :

wmax = solve A ' w = 0, w , lmax =5000K 3$

25003

6, Amax = A

25003

;

wmax = solve A ' w = 0., w , lmax =5000.K 3$

25003

6, Amax = A

2500.3

;

Question4Version1Constraint := 3 wC 6 l = 5000

Objective =12

w 5000K 3 w

wmax = 10 403 , K10 403 , lmax =1250

3, Amax =

6612725

wmax = 200.7485990, K200.7485990 , lmax = 416.6666667, Amax = 2645.080000

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Question4Version2; Constraintd 3$wC 6$l = 6000;

A w d3$w$ 6000K 3$w

6:


wmax = solve A ' w = 0, w , lmax =6000K 3$1000

6, Amax = A 1000 ;

wmax = solve A ' w = 0., w , lmax =6000.K 3$1000

6, Amax = A 1000 ;

Question4Version2Constraint := 3 wC 6 l = 6000

Objective =12

w 6000K 3 w

wmax = 1000, lmax = 500, Amax = 1500000wmax = 1000., lmax = 500.0000000, Amax = 1500000

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> > Question5Version1; Constraintd 3$l$w = 62450;

A w d 6$624503 $w

C 3$w :


wmin = solve A ' w = 0, w , lmin =62450

3$103

3747, Amin = A

103

3747 ;

wmin = solve A ' w = 0., w , lmin =62450.

3$103

3747., Amin = A

103

3747. ;

Question5Version1Constraint := 3 l w = 62450

Objective =124900w

C 3 w

wmin =103

3747 , K103

3747 , lmin =53

3747 , Amin = 20 3747

wmin = 204.0424792, K204.0424792 , lmin = 102.0212396, Amin = 1224.254875

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Question5Version2; Constraintd 3$l$w = 60450;

A w d 6$604503 $w

C 3$w :


wmin = solve A ' w = 0, w , lmin =60450

3$10 403, Amin = A 10 403 ;

wmin = solve A ' w = 0., w , lmin =60450.

3$10 403., Amin = A 10 403. ;

Question5Version2Constraint := 3 l w = 60450

Objective =120900w

C 3 w

wmin = 10 403 , K10 403 , lmin = 5 403 , Amin = 60 403wmin = 200.7485990, K200.7485990 , lmin = 100.3742995, Amin = 1204.491594

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Question6Version1; Constraintd x$y = 1275;

C x d 26$ 104 C 2$xC 2$2$1275x

:

Objective = C x ; plot C x , x = 0 ..100 :

xmin = solve C ' x = 0, x , ymin =1275

5 102, Cmin = C 5 102 ;

xmin = solve C ' x = 0., x , ymin =1275.

5 102., Cmin = C 5 102. ;

Question6Version1Constraint := x y = 1275

Objective = 2704C 2 xC5100x

xmin = 5 102 , K5 102 , ymin =52

102 , Cmin = 2704C 20 102

xmin = 50.49752469, K50.49752469 , ymin = 25.24876234, Cmin = 2905.990098

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> > Question6Version2; Constraintd x$y = 1075;

C x d 20$ 104 C 3$xC 3$2$1075x

:

Objective = C x ; plot C x , x = 0 ..100 :

xmin = solve C ' x = 0, x , ymin =1075

5 86, Cmin = C 5 86 ;

xmin = solve C ' x = 0., x , ymin =1075.

5 86., Cmin = C 5 86. ;

Question6Version2Constraint := x y = 1075

Objective = 2080C 3 xC6450x

xmin = 5 86 , K5 86 , ymin =52

86 , Cmin = 2080C 30 86

xmin = 46.36809248, K46.36809248 , ymin = 23.18404624, Cmin = 2358.208554

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Question7Version1; Constraintd sqrt 152 C 25K h 2 ;

C h d 5$ 2$50$h C 5$ 2$30$h C 5$ 2$12$30$ 25K h C 14$ 2$50$sqrt 152

C 25K h 2 : Objective = C h ; plot C h , h = 0 ..40 :

hmin = solve C ' h = 0, h , lmin = sqrt 152 C 25K 25K1341

6152

, Cmin = C 25

K1341

615 ;

hmin = solve C ' h = 0., h , lmin = sqrt 152 C 25K 25K1341

615.2

, Cmin

= C 25K1341

615. ;

Question7Version1

Constraint := 225C 25K h 2

Objective = 650 hC 3750C 1400 225C 25K h 2

hmin = 25K1341

615 , lmin =2841

615 , Cmin = 20000C 750 615

hmin = 17.13684107, lmin = 16.93603461, Cmin = 38599.39515

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Question7Version2; Constraintd sqrt 152 C 25K h 2 ;

C h d 7$ 2$50$h C 5$ 2$30$h C 5$ 2$12$30$ 25K h C 16$ 2$50$sqrt 152

C 25K h 2 : Objective = C h ; plot C h , h = 0 ..40 :

hmin = solve C ' h = 0, h , lmin = sqrt 152 C 25K 25K177

152

, Cmin = C 25

K177

15 ;

hmin = solve C ' h = 0., h , lmin = sqrt 152 C 25K 25K177

15.2

, Cmin = C 25

K177

15. ;

Question7Version2

Constraint := 225C 25K h 2

Objective = 850 hC 3750C 1600 225C 25K h 2

hmin = 25K177

15 , lmin =327

15 , Cmin = 25000C 5250 15

hmin = 15.59418330, lmin = 17.70506673, Cmin = 45333.16257

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Documents

mathsci2.appstate.edumathsci2.appstate.edu/~osbornejm/MAT1030/QuizzesHomework...MAT$1030$$Exam$4$Version$1:$ $ Last$Name$_____ $,$First$Name_____ $ $ Circle$Your$SecAon$Number:$$$Sec.$112$(3F350)$$$or$Sec.$114$(4F450)$$$

mathsci2.appstate.edumathsci2.appstate.edu/~osbornejm/MAT1030/QuizzesHomework...MAT$1030$$Exam$4$Version$1:$ $ Last$Name$_ $,$First$Name_ $ $ Circle$Your$SecAon$Number:$$$Sec.$112$(3F350)$$$or$Sec.$114$(4F450)$$$