Question 12 Your answer is INCORRECT Choose only onc of problem 12 or 13. This i
ID: 2891718 • Letter: Q
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Question 12 Your answer is INCORRECT Choose only onc of problem 12 or 13. This is a w ritten question, worth 13 poinls. DO NOT place the problem code on the answer sheet. A proctor wil fill this out after exam submission. Show all steps (work) on your answer sheet for full credit. Problem Code: 1277 Four congruent squares are to be cut from the cormers of a rectangular picce of cardboard 15 inches long by 8 inches wide. The remaining cross like piece is then to he fokded into an open box t1) Express the volume of the box as a function of one variable and give the domain of the Find the dimensions of the box that givc the maximum volume aIhave placed my work and my answer on my answer sheet b) I want to have points deducted from my test for not working this problem. Question 13 Your answer is CORRECT Choose only one of problem 12 or 13. This is a written question, worth 13 points. DO NOT place the prohiem code on the answer sheet. A proctor willfl this out after exam submission. Show all steps (work) on your answer sheet for full credit Problem Code: 1365 A rectangle has its base on the x-axis, its hottom left corner on the origin and its top right coner on the line ti) Express the area of the rectangle as a function of one variable and give the domain of the function, (n) What is the naxintum area of the rectangle? a)have placed my work and my answer on my answer sheet I want lo have P"ntsaleducted from my les·fur not working this problem.Explanation / Answer
12)
(i)
dimensions of opentop box are x , 15-2x , 8-2x
x>0, 15-2x>0, 8-2x>0
=>x>0, x<7.5,x<4
=>0<x<4
volume of the opentop box,V= x(15-2x)(8-2x) in3, domain is 0<x<4 or x=(0,4)
(ii)
V= (120x-46x2+4x3)
dV/dx= (120-92x+12x2),d2V/dx2= (-92+24x),
when volume is maximum dV/dx= 0,d2V/dx2<0
=>(120-92x+12x2)=0
=>x2-(23/3)x +10=0
by quadratic formula
x=5/3 , x=6
but only x=5/3 lies in domain
d2V/dx2= (-92+24*(5/3))
=>d2V/dx2= -52
=>d2V/dx2<0
dimensions of opentop box that gives maximum volume are (5/3) , 15-2(5/3) , 8-2(5/3)
dimensions of opentop box that gives maximum volume are (5/3) , (35/3) , (14/3)
dimensions of opentop box that gives maximum volume are 5/3 in , 35/3 in ,14/3 in exactly
dimensions of opentop box that gives maximum volume are 1.67 in , 11.67 in ,4.67 in approximately
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13)
(i)
area of rectangle ,A=xy
area of rectangle ,A=x(-4x+13)
area of rectangle ,A=-4x2+13x
for area , x>0,y>0
=>x>0, -4x+13>0
=>x>0,x<13/4
domain is 0<x<13/4 or 0<x<3.25 or x=(0,3.25)
(ii)
A=-4x2+13x
dA/dx=-8x+13,d2A/dx2=-8
for maximum area dA/dx=0,d2A/dx2<0
=>-8x+13=0
=>x=13/8
=>x=1.625
maximum area of rectangle ,A=-4*1.6252+(13*1.625)
maximum area of rectangle ,A=10.5625 or 169/16
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