Math quick question

You have 500 cm² of material available to cover different shapes. what is the maximum volume of each object that you could make with this amount of material?
A.) a cube
B.) a sphere

I've been stuck on this for so long, please show me how to do this step by step, thanks :)

  1. 👍 0
  2. 👎 0
  3. 👁 75
asked by Miley
  1. So you're given the limitation of 500 cm^2. This is essentially the surface area of the different solids if you choose to form them into that shape.

    The best way to calculate the maximum volume with a fixed surface area is to find the ratio of volume to surface area. The solid with a larger ratio is is able to hold more volume per surface area, which naturally means it can hold more volume with a fixed surface rea.

    First the cube:

    Surface area of the cube, you need to add the area of all 6 sides. For simplicity sakes, we assume that the length of the cube is variable x. So the surface area is equal to 6*(x^2). The volume is much easier calculate, x^3.

    The ratio between volume to surface area of a cube is x^3/(6*x^2). Cancel out a x^2 from the top and bottom and you get x/6.

    Now the sphere:

    Once again, we assume for simplicity sakes that the length of the radius is x. If you still don't understand this, the reason we do this is because if we choose to set a ratio comparing the ratios of sphere and cubes, the x variable cancels.

    The surface area for a sphere is 4πx^2. The volume is 4/3πx^3. Set a ratio of volume to surface area. 4/3πx^3/4πx^2 which simplifies to x/3.

    Now compare the different ratios.
    x/6 and x/3. As you can see, the ratio of volume to surface area is much bigger for the sphere than the cube. Therefore forming a sphere is the best way to maximize volume. It's the same logic with 2D shapes. A circle can hold more area with a set perimeter than a square.

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math urgent help please!!

    You have 500 cm² of material available to cover different shapes. what is the maximum volume of each object that you could make with this amount of material? A.) a cube B.) a sphere I've been stuck on this for so long, please

    asked by Miley on January 7, 2014
  2. Math

    The internal and external radii are estimated to be 6 cm and 8 cm respectively, to the nearest whole number. The height of the cylinder is exactly 14 cm. (a) Determine the exact values for internal and external radii which will

    asked by Kd on November 7, 2018
  3. AP Calculus

    If 600 cm2 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. Answer to the nearest cubic centimeter without commas. For example, if the answer is

    asked by Alyssa on October 16, 2014
  4. calculus

    If 600 cm2 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. Answer to the nearest cubic centimeter without commas. For example, if the answer is

    asked by jake on January 19, 2015
  5. Calculus

    An open top bos with a square base is to be constructed from 192 sq ft of material. What should be the dimensions of the box if the volume is to be a maximum? What is the maximum volume?

    asked by Derek on October 29, 2012
  6. Math

    Q) 'What are the dimensions of a cylinder that has the same maximum volume as the box (1131.97 cm2), but uses the minimum amount of material to make it?' Here are the notes I have so far: SA=2Pi r^2 + 2 Pi r (1131.97/Pi r^2) Now i

    asked by Josh on November 9, 2007
  7. Geography

    How does erosion shape mountains? A. It creates pressures in Earth's crust that causeto bend B. I removes material from mountains, changing their shapes. C. By breaking down material on Earth's surface, it changes mountains'

    asked by Bob Bob Chocolat on August 29, 2019
  8. Living environment

    The apperance of the greatest percentageof white mice occured (1) before the maximum number of days of snow cover (2)at the same time as the maximum number of days of snow cover (3) after the maximum number of days of snow cover

    asked by lisa on January 3, 2012
  9. calculus

    A rectangular box is to be constructed from two different materials. The box will have a square base and open top. The material for the bottom costs $4.25/m2. The material for the sides costs $2.50/m2. Find the dimensions of the

    asked by j on April 9, 2011
  10. art

    this image was created by using basic math concepts that include all of the following except A)knowing how to create a pattern B)using understanding the angles of the shapes C) a basic understanding of different geometric shapes.

    asked by tom on October 5, 2016

More Similar Questions