A silo (base not included) is to be constructed in the form of a cylinder surmou

Question

A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is 3 times as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed at 2000 cubic units and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction. The radius of the cylindrical base (and of the hemisphere) is The height of the cylindrical base is

Explanation / Answer

let radius of cylindrical base = radius of hemispere = r

let height of cylinder =h

volume of silo =volume of cylinder +volume of hemisphere

=>r2h+(2/3)r3 =2000

=>h =[2000-(2/3)r3]/r2

let cost of construction of cylinder =x per square unit => cost of construction of hemisphere =3x per square unit

total surface area =2rh+ 2r2

total cost of construction C=(2rh)x+ (2r2)3x

C=x[2rh+ 6r2]

C=x[2r([2000-(2/3)r3]/r2) + 6r2]

C=x[(4000/r)-(4/3)r2 + 6r2]

C=x[(4000/r)+(14/3)r2]

for minimum cost of construction dC/dr =0 , d2C/dr2 >0

dC/dr =x[(-4000/r2)+(28/3)r]

x[(-4000/r2)+(28/3)r]=0

[(-4000/r2)+(28/3)r]=0

(28/3)r=(4000/r2)

r3=4000*3/(28*)

r=5.2 ft

d2C/dr2 =x[(8000/r3)+(28/3)]

at r =5.2 ,d2C/dr2>0

r =5.2 ,h =[2000-(2/3)r3]/r2

h =[2000-(2/3)5.23]/*5.22

h =20.6 ft

radius of cylindrical base (and of the hemisphere) is 5.2 ft

height of cylindrical base is 20.6ft

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

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