Dimensions of box with largest volume

In summary, the conversation is about finding the dimensions of a rectangular box with the largest volume given a surface area of 64cm^2. The person first rearranges the formula for surface area to solve for z and then takes the partial derivatives of z with respect to x and y. They set these derivatives equal to zero to find the critical point but are unable to find a real root for y. They clarify that they are maximizing the volume of the box and not just one dimension. The other person suggests finding the maximum volume of a box with surface area 64cm^2 first, and the original person agrees to continue solving from there.
  • #1
evilpostingmong
339
0

Homework Statement


Find the dimensions of a rectangular box with the largest volume with surface area 64cm^2.



Homework Equations


area of a rectangular prism 2xy+2yz+2xz


The Attempt at a Solution


took 2xy+2yz+2xz=64
rearranging for z:
z=(xy-32)/(y+x)
partial derivative of z with respect to x
y/(y+x)-(xy-32)/(y+x)^2
partial derivative of z with respect to y
x/(y+x) -(xy-32)/(y+x)^2
setting both derivatives equal to zero to obtain the critical point...
Here's where I hit a wall-
I cannot get a real root for y.
y/(y+x)-(xy-32)/(y+x)^2
y^2+xy=xy-32
y^2=-32
Please only help me up to this point until I ask you to help me further, I
want to challenge myself. Thank you!
 
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  • #2
what are you maximizing?
 
  • #3
The volume of this box.
 
  • #4
you're maximizing xyz?
 
  • #5
Oh wait I'm finding the maximum dimensions that can be used to get the surface area of 64cm^2. So I maximize 2xy+2yz+2xz=64
 
  • #6
'maximum dimensions'?
 
  • #7
The problem states that I must find the dimensions of a rectangular box with the largest volume if it's total surface area is 64cm^2. What we don't know is how long those dimensions are.
 
  • #8
first you should find the maximum volume of a box with surface area 64cm^2.
 
  • #9
Okay! I'll take the problem from there and tell you how I made out.
 

1. How do you find the dimensions of the box with the largest volume?

The dimensions of the box with the largest volume can be found by using the formula V = lwh, where V is volume, l is length, w is width, and h is height. To find the dimensions with the largest volume, you can use calculus to take the derivative of the volume formula and set it equal to zero. This will give you the critical points, which can then be evaluated to determine the dimensions of the box with the largest volume.

2. Can the dimensions of the box with the largest volume be negative?

No, the dimensions of the box with the largest volume cannot be negative. In order for a box to have a volume, all of its dimensions must be positive. If the dimensions were negative, the box would have no physical meaning and therefore cannot have a volume.

3. How do you know if the box with the largest volume is a cube or a rectangular prism?

The box with the largest volume can be either a cube or a rectangular prism depending on the constraints given. If the constraints state that all sides of the box must be equal, then the box with the largest volume will be a cube. However, if there are no constraints on the dimensions, then the box with the largest volume will be a rectangular prism with different length, width, and height values.

4. Are there any real-life applications for finding the dimensions of the box with the largest volume?

Yes, there are many real-life applications for finding the dimensions of the box with the largest volume. For example, in manufacturing, companies often need to determine the dimensions of a box that can hold the maximum amount of a product to optimize storage and transportation. In architecture, finding the dimensions of a room with the largest volume can help maximize usable space. Additionally, in engineering, determining the dimensions of a container with the largest volume can help improve efficiency in transporting liquids or gases.

5. Can the dimensions of the box with the largest volume change depending on the units used for measurement?

Yes, the dimensions of the box with the largest volume can change depending on the units used for measurement. For example, if the dimensions are measured in inches, the resulting volume will be in cubic inches. However, if the dimensions are measured in centimeters, the resulting volume will be in cubic centimeters. It is important to be consistent with units when finding the dimensions of the box with the largest volume to ensure accurate results.

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