Calc

A rectangular storage container with an open top is to have a volume of 15m^3. The length of the base is twice the width of the base. Material for the base costs $6 per square meter. Material for the sides costs $7 per square meter. Find the cost of the materials for the cheapest such container. Round answer to nearest hundredth.

I know that you need to find an equation, get the derivative, find the critical numbers, and then find the minimum. But, I have no clue on how to find the equation! If someone could please help me!

You will need a diagram.
Visualize the box "flattened", you have a rectangular base, with rectangles on each of its sides.
These four rectangles must have the same width, namely the height of the box.
Let that height be h m.
Let the width of the base be x m, then its length is 2x m.

So the volume is x(2x)h or 2hx^2 m^3
but we know this is 15.
So one of the equations is 2hx^2 = 15 or
h = 7.5/x^2

Since you want to minimize the Cost, you now need an equation for Cost

Cost of base = 6(x)(2x)=12x^2
Cost of sides = 2(7)(x)h) + 2(7)(2x)(h)
=42xh

Cost = 12x^2 + 42xh
=12x^2 + 42x(7.5/x^2)
=12x^2 + 315/x

Cost' = 24x - 315/x^2 = 0 for a min of Cost

Solve this....i got x=2.3588....

You actually have to plug it into the original cost equation and I get $200.31 and it's the right answer! Thank you so very much for your help!!!!

  1. 👍 0
  2. 👎 0
  3. 👁 248

Respond to this Question

First Name

Your Response

Similar Questions

  1. Calculus

    A rectangular tank with a square base, an open top, and a volume of 864 ft^3 is to be constructed of the sheet steel. Find the dimensions of the tank that minimize the surface area

    asked by Ashlyn on April 22, 2018
  2. Math

    A rectangular piece of cardboard measuring 12 cm by 18 cm is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each square in

    asked by NR on March 23, 2017
  3. Math

    A trash company is designing an​ open-top, rectangular container that will have a volume of 135 ft^3.The cost of making the bottom of the container is​ $5 per square​ foot, and the cost of the sides is​ $4 per square foot.

    asked by Todd on October 8, 2016
  4. Calculus (Optimization)

    A rectangular piece of cardboard, 8 inches by 14 inches, is used to make an open top box by cutting out a small square from each corner and bending up the sides. What size square should be cut from each corner for the box to have

    asked by Mishaka on December 16, 2011
  5. Geometry

    On a rectangular piece of cardboard with perimeter 11 inches, three parallel and equally spaced creases are made. The cardboard is then folded along the creases to make a rectangular box with open ends. Letting x represent the

    asked by Valerie on January 21, 2011
  1. maths-urgently needed

    The volume of a cylinder is 48.125 cm3, which is formed by rolling a rectangular paper sheet along the length of the paper. If cuboidal box (without any lid i.e., open at the top) is made from the same sheet of paper by cutting

    asked by Anonymous on January 16, 2013
  2. Pre Cal 12

    A 12cm by 8cm rectangular piece of metal is to be made into an open-top box by cutting a sqaure from corner and folding up the resulting flaps (sides). If the volume of the lidless box is 36 cm what are the integer dimensions of

    asked by Aaron on January 26, 2017
  3. Calculus

    A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $9 per m2. Material for the sides costs $150 per m2. Find the dimensions of the

    asked by Emily on November 26, 2017
  4. Math

    A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in by 12 in by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the

    asked by KC on September 10, 2012
  5. calculus

    Find the dimensions of the largest open-top storage bin with a square base and vertical sides that can be made from 108ft^2 of sheet steel. (Neglect the thickness of the steel and assume that there is no waste)

    asked by Barbara on January 3, 2009
  6. Math

    Bob has designed a rectangular storage unit to hold large factory equipment. His scale model has dimensions 1 m by 2 m by 4 m. By what amount should he increase each dimension to produce an actual storage unit that is 9 times the

    asked by Max on April 6, 2015

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