Ask your own question, for FREE!
Mathematics 74 Online
OpenStudy (anonymous):

An open rectangular box with square base is to be made from 48 ft^2 of material. What dimensions will result in a box with the largest possible volume?

OpenStudy (bahrom7893):

bear with me.. solving

OpenStudy (bahrom7893):

Surface area is area of the bottom: x^2 + 4 sides: 4*xy Surface area = 48ft^2 = x^2 + 4xy Volume is x^2*y: V = x^2 * y

OpenStudy (bahrom7893):

Solve for y in the surface area: 48 = x^2 + 4xy 48-x^2 = 4xy (48-x^2)/4x = y

OpenStudy (bahrom7893):

y = (48 - x^2) / 4x V = x^2 * y = x^2 * (48 - x^2) / 4x = x ( 48-x^2) / 4

OpenStudy (bahrom7893):

Simplify a bit: V = 48x/4 - x^3/4 = 12x - x^3/4 Take the derivative and set it equal to 0: V ' = 12 - 3x^2 / 4 = 0

OpenStudy (bahrom7893):

Solve the equation: 12 - 3x^2 / 4 = 0 -3x^2/4 = -12 x^2/4 = 4 x^2 = 4*4 = 16

OpenStudy (bahrom7893):

x = + or - 4, but in this case it's a plus because the edge of the box cannot be negative.. anyway let's double check if it's min or max. To do so, take the 2nd derivative: V" = -6x / 4 = - 3x / 2. Which is positive if x = -4 and negative if x = 4. Maximum occurs when 2nd derivative is negative therefore x = 4 is max

OpenStudy (bahrom7893):

Okay so now that you have x = 4, find y: y = (48 - x^2) / 4x y = (48 - 16) / 16 = 32/16 = 2 So dimensions are: The sides of the base must be equal to 4, and the height of the box must be 2. In which case the volume = area of the base * height: V = 4^2 * 2 = 16 * 2 = 32

OpenStudy (bahrom7893):

Hope this helped =)

OpenStudy (bahrom7893):

by the way units are: x = 4ft y = 2ft V = 32ft^3

Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!

Join our real-time social learning platform and learn together with your friends!
Latest Questions
SanestTf2Fan: do you guys hear the flibbity jibbity jibber jabber????
1 minute ago 6 Replies 1 Medal
MakaylaChuck23: I love Donald J. Trump.
5 seconds ago 21 Replies 1 Medal
MakaylaChuck23: rate ud83dude00
7 minutes ago 33 Replies 1 Medal
roy94: @everyone im gay.
7 minutes ago 8 Replies 2 Medals
idontexistrn1: anybody have some music suggestions?
9 minutes ago 44 Replies 2 Medals
Ash2746: This is what I did in class
31 minutes ago 61 Replies 3 Medals
Breathless: WOmp
1 hour ago 18 Replies 1 Medal
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!

Join our real-time social learning platform and learn together with your friends!