Solv. Calculus Problems: Max Vol, US Mail Stipulations

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In summary, the first problem involves finding the radius and height of a cylindrical box with the largest possible volume, using a given length of ribbon. The second problem involves finding the dimensions of a box with maximum volume that can be sent through the U.S. mail, based on length and girth restrictions. The solution to the first problem is a height of 10cm and radius of 10cm, while the second problem results in a width and height of 18in and length of 36in. The method of tying the ribbon is similar to tying a birthday cake.
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rain
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I have some calculus problems that I don't know if I did it right.

1)A cylindrical box will be tied up with ribbon. The longest piece of ribbon available is 130cm long, and 10cm of that are required for the bow. Find the radius and height of the box with the largest possible volume.
-my answer is h=10cm and r=10cm

2)The U.S. Postal Service stipulates that any boxes sent through the mail must have a length plus girth totaling no more than 108in. Find the dimensions of the box with maximum volume that can be sent through the U.S. mail, assuming that the width and the height of the box are equal.
-my answer is width/height=18in and length=36in

Please check my answer if I got it right, if not, I'll post my steps for you guys to check. Thanks in advance.
 
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  • #2
How do you tie the ribbon?
 
  • #3
hm...like you would tie a birthday cake...a cross on top and bottom, 4 strips on the side...do you get what I mean?
 
  • #4
Everything looks right.
 
  • #5
thanks a lot. people are nice in here.
 

FAQ: Solv. Calculus Problems: Max Vol, US Mail Stipulations

What is the maximum volume that can be achieved for a given surface area?

The maximum volume can be achieved by using a shape with maximum curvature, such as a sphere or a cylinder, which has the most efficient packing of volume within a given surface area.

How can calculus be used to solve these types of problems?

Calculus is used to find the critical points of the function representing the volume, and then the second derivative test is used to determine if the point is a maximum or minimum. This allows us to find the maximum volume for a given set of constraints.

Can the same method be applied to different types of optimization problems?

Yes, the same method can be applied to any type of optimization problem in which a function needs to be maximized or minimized subject to certain constraints. Calculus is a powerful tool for solving these types of problems.

Are there any real-world applications of solving calculus problems for maximum volume?

Yes, this type of problem is commonly used in engineering and architecture to determine the most efficient shape for a given structure. It is also used in economics and business to optimize production and profits.

What are some common stipulations when solving calculus problems for maximum volume?

Some common stipulations include constraints on the dimensions or shape of the object, limitations on the materials used, and restrictions on the allowed techniques for finding the maximum volume. These stipulations often reflect real-world limitations and must be taken into account when solving the problem.

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