Problem 1:) Find three numbers x; y; and z, such that the sum of the numbers is
ID: 3343867 • Letter: P
Question
Problem 1:) Find three numbers x; y; and z, such that the sum of the numbers is 30 and the
sum of the squares of the numbers is minimum. For example, %uDBC0%uDC002; 20; 12 are three numbers whose
sum is 30 and the sum of their squares is (%uDBC0%uDC002)2 +(20)2 +(12)2 = 4+400+144 = 549. Another
possibility is 20; 5; 5 and the sum of those squares is 450, which is smaller than the previous
example. Find three numbers such that the sum of the squares is the smallest number possible.
Problem 2:) Find the dimensions of the right circular cylinder of volume V0 which has the
smallest surface area.
Problem 3:) Consider a closed retangular box of volume 100 square foot. Suppose the material
to construct the top and bottom of the box is 10 dollars per square foot, and the material cost
for the sides is 5 dollars per square foot. What is the dimensions of the cheapest such box?
Please show me all the steps the solve these three problems and it will help me out a lot
Explanation / Answer
1. x+y+z = 30
Square :
x^2 + y^2 + z^2 = 900 - 2(xy+yz+zx)
Minimum for symmetry:So x = y =z = 10
Hence min sum of squares = 300
2. pi . r^2 . h = 100
2(pi)rh + pi.r^2 = minimum
r^2 + 2rh = min.
r^2 + 100/pi.r = min
differentiate:
2r - 100/pi.r^2 = 0
r = (50/pi)^(1/3)
h = 100/pi.(pi/50)^(2/3)
3. l and b are equal by symmetry.
l^2.h = 100
minimise : 10. 2. l^2 + 5.4.l.h
minimise : 20l^2 + 20 lh
minimise : l^2 + lh
h = 100/l^2
differentiate
2l -100/l^2 = 0
l = 50^1/3
b = 50^1/3
h = 100/(50)^(2/3)