Edge Rivet

Posted May 8, 2009 By admin

CALCULUS: volume of open box?

An open 100cm^3 rectangular box is equally riveted along its edges (top edges excluded). If the length of the box is twice as long as its breadth find its dimensions so that the cost of the rivets is cheapest.

I know the volume should be length x breadth x height but i really don't know how to get the equation to derive The answers are: b=4.05, l=8.1, h=3.04

thank you

Hello

Length = a
breadth = b
height = c
b = 1/2 a (box is twice as long as broad)

volume = a*b*c = a*1/2a*c = 1/2a^2*c = 100 cm^3 --->
c = 200/a^2

The length of the riveted edge is 2a + 2b + 4c
L = 2a + 2(1/2a) + 4(200/a^2)
L = 2a + a + 800/a^2
L = 3a + 800*a^-2

To find the minimum of the length, you have to set the first derivative = 0:
L' = 3 +(-2)*800*a^-3
L' = 3 - 1600*a^-3 = 0
1600/a^3 = 3
1600 = 3*a^3
1600/3 = a^3
a = 8,1 cm

a = 2b
8,1 = 2b
b = 4,05 cm

a*b*c = 100
c = 100/(8,1*4,05)
c = 3,04

Regards