Question

A closed rectangular box with a volume of is to be made of three different materials. The cost of the material for the top and the bottom is $9 per sq. ft, the cost of the material for the front and the back is $8 per sq. ft. and the cost of the material for other two sides is $6 per sq. ft. Find the dimensions of the box (by two ways if possible) so that the cost of materials is minimized.​

Answer 1

Answer:

Let the dimension of the box with square base be x,x,h

Volume of the rectangular box V=x

2

h

x

2

h=1000

Area of top =x

2

Area of base =x

2

Area of faces =4xh

E=15x

2

+25x

2

+20(4xh)+300

E=40x

2

+80x(

x

2

1000

)+300

=40x

2

+

x

80×1000

+300

For maxima or minima,

dx

dE

=0

∴80x−

x

2

80×1000

=0

∴x

3

=1000

x=10

dx

2

d

2

E

=80+

x

3

2×80×1000

=+ive

∴ Min. when x=10

h=

x

2

1000

=10

Hence the box should be a cube of edge 10 feet.