Math

If 1,200 cm^2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters. Answer to the nearest cubic centimeter.

  1. 👍 0
  2. 👎 0
  3. 👁 283
  1. You want a square base box with a fixed surface area:

    SA = s² + 4sh
    1200 = s² + 4sh

    What's the maximum volume you can make with that box?

    V = s²h

    Solve the first equation for h, then substitute and solve for s into the volume equation:

    1200 = s² + 4sh
    -4sh = s² - 1200
    h = -s/4 + 300/s

    V = s²h
    V = s²(-s/4 + 300/s)
    V = -s³/4 + 300s

    Now that we have volume in terms of one variable, we can find its maximum by taking the first derivative of that function, set it to zero, then solve for s:

    dV/ds = -(3/4)s² + 300
    0 = -(3/4)s² + 300
    (3/4)s² = 300
    s² = 400
    s = 20

    (throwing out the negative value of "s" since we're dealing with a box)

    Now that we have "s", substitute it back into the volume equation to determine what that volume is:

    V = -s³/4 + 300s
    V = -(20)³/4 + 300(20)
    V = -8000/4 + 6000
    V = -2000 + 6000
    V = 4000

    The maximum volume is 4000 cm³ when the side of the box is 20cm² and the height is 10 cm.

    V = s²h
    4000 = 20²h
    4000 = 400h
    h = 10

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    If 1600 square centimeters of metal is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume = ?

    asked by Anonymous on February 17, 2014
  2. Calculus Please Help

    If 1100 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. I'm supposed to type the answer in cubic centimeters... my answer was 2500 cm^3 but

    asked by Eva on April 6, 2011
  3. Math!!

    If 1800 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box Volume=____________ I did sqrt(1800)=42.4264 and then 42.4264/3= 14.142 cm then

    asked by Anonymous on November 29, 2009
  4. calculus

    A square-based, box-shaped shipping crate is designed to have a volume of 16ft^3. The material used to make the base costs twice as much (per ft^2) as the material in the sides, and the material used to make the top costs half as

    asked by Bob on November 4, 2010
  5. calculus

    If 2100 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume = cubic centimeters.

    asked by kwack on September 7, 2010
  1. Math

    The material used to make a storage box costs $1.25 per square foot. The boxes have the same volume. How much does a company save by choosing to make 50 of Box 2 instead of 50 of Box 1?

    asked by Yasmine on May 10, 2017
  2. Calculus

    A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount of material used.

    asked by Anna on January 28, 2014
  3. calculus

    A container company is tasked to make an open-top rectangular box with a square base. The box must have a volume of 108cm^(3). let the length of the sides of the square base be x cm and the height h cm. (1) what value of x will

    asked by NIKI on August 4, 2011
  4. Math (calc)

    An open box with a square base is to have a volume of 12ft^3. Find the box dimensions that minimize the amount of material used. (round to two decimal places). it asks for the side length and the height. Please help asap due in a

    asked by Kyle on February 8, 2017
  5. calculus

    A rectangular closed box with a square base is to have a capacity of 27 cubic inches determine the least amount of material required.

    asked by gillian on April 27, 2016
  6. math

    Would the answer be 2000 to this? with 1,200 i got 4,000? If 600cm^2 of material is available to make a box with a square base and a closed top, find the maximum volume of the box in cubic centimeters.

    asked by Gary on March 6, 2015

You can view more similar questions or ask a new question.