Optimization problems. These can be solved by translating the problem into math

Question

Optimization problems. These can be solved by translating the problem into math language (equations) and then finding the minimum or maximum of the function. Find two positive numbers whose sum is 400 and whose product is a maximum. Find x and why so that x + 2y = 50 and (x - 1)(y + 2) is maximum. A beverage company wants to minimize the material used for producing a soda can. The soda can needs to be cylindrical and hold 12 fl oz (354 ml) of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction. Use the variables "r" for the radius and "h" for the height of the cylinder.

Explanation / Answer

a) let umbers are x,y

sum is 400

x+y =400

y =400-x

product P=xy

P=x(400-x)

P=-(-400x+x2)

P=-(-2002+2002-400x+x2)

P=2002-(2002-400x+x2)

P=2002-(x-200)2

P=40000-(x-200)2

vertex is (200,40000)

product is maximum when x =200

y =400-x

y =400-200

y =200

2 positive numbers are 200,200

---------------------------------

b) given x+2y =50

=>x=50-2y

let unction is f=(x-1)(y+2)

f=(50-2y-1)(y+2)

f=(49-2y)(y+2)

f=49y-4y-2y2+98

f=98+45y-2y2

f has maximum when df/dy=0 ,d2f/dy2<0

df/dy=45-4y ,d2f/dy2=-4<0

45-4y=0

y=45/4

y=11.25

x=50-2y

x=50-2(11.25)

x=50-22.5

x=27.5

x is 27.5 for (x-1)(y+2) to be maximum

--------------------------------------------

c)radius =r , height =h

volume v =r2h

given volume =354 ml=354 cm3

r2h=354

h=354/(r2)

amount of material used will be minimum when surface area is minimum

surface area of can s=2r2+2rh

s=2r2+2r(354/(r2))

s=2r2+(708/r)

for minimum surface area ds/dr =0, d2s/dr2>0

ds/dr=4r-(708/r2), d2s/dr2=4 +(1416/r3)

4r-(708/r2)=0

r3=708/(4)

r=3.8336cm

d2s/dr2=4 +(1416/3.83363) >0

so surface area is minimum when r=3.8336cm

h=354/(*3.83362) =7.6672cm

so h =7.6672cm , r =3.8336cm

Related Questions

Navigate

• Browse (All)
• Browse O
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.