Optimizing a Container’s Shape: Sketching, Volume, and Surface Area

When designing a container, it's important to consider both the volume it holds and the amount of material needed to construct it. This includes understanding the shape and dimensions that achieve the best balance between these factors. In this article, we will explore a specific scenario where a container has a circular base and a volume of 1000 cm3. We'll delve into the steps to sketch the container, calculate its volume, and determine the minimum surface area necessary for a closed lid.

1. Sketching the Container

The first step in solving this problem involves sketching the containerrsquo;s dimensions. Letrsquo;s start by assuming the container has a circular base and a height. Given the volume, we need to find the radius and height that satisfy the volume requirements.

2. Calculating the Volume

The formula for the volume of a cylinder is given by:

Volume (V) πr2h

Where:

πr2 is the area of the base, h is the height of the cylinder.

To find the radius (r) and height (h) when the volume is 1000 cm3, we can rearrange the formula:

h V / (πr2)

3. Determining the Minimum Surface Area

The surface area (S) of a closed cylinder is given by the formula:

Surface Area (S) 2πrh 2πr2

This can be simplified to:

S 2πrh 2πr2

We need to find the dimensions that minimize the surface area while maintaining the given volume. To do this, we can take the first differential of the surface area with respect to the radius (r) and set it to zero. This will help us find the critical points.

3.1. First Derivative and Critical Points

dS/dr 2πh 4πr 0

Rearranging the equation, we get:

2πh -4πr

Dividing by 2π, we have:

h -2r

Since height cannot be negative, we need to re-evaluate our approach. Instead, letrsquo;s use the total volume to express h in terms of r:

h V / (πr2) 1000 / (πr2)

Substituting this into the surface area formula:

S 2πr(1000 / (πr2)) 2πr2 2000 / r 2πr2

3.2. Second Derivative to Determine Minimum

To confirm that this is a minimum, we take the second derivative of S with respect to r:

d2S/dr2 2000 / r3

Since the second derivative is always positive for positive r, this confirms that h 2r is a minimum.

4. Calculating the Radius and Surface Area

Using the expression for h 2r, we can substitute into the volume formula:

1000 πr2(2r) 2πr3

Solving for r:

2r3 1000

r3 500

r 5001/3 ≈ 7.937 cm

Now, substituting back to find h:

h 2r 2(7.937) ≈ 15.874 cm

The surface area (S) is then calculated as:

S 2πrh 2πr2 2π(7.937)(15.874) 2π(7.937)2

S ≈ 619.8 cm2

Conclusion

The optimal dimensions for a closed cylindrical container with a volume of 1000 cm3 are approximately a radius of 7.937 cm and a height of 15.874 cm, resulting in a surface area of about 619.8 cm2. This approach ensures that the container is both functional and efficient in its material usage.

Related Keywords

circular base

A circular base is an essential feature in many cylindrical containers, providing structural integrity and a consistent shape throughout the object.

volume calculation

Calculating the volume of a container is crucial for ensuring it meets the required capacity. The formula for a cylinder's volume is straightforward and can be applied to various design needs.

minimum surface area

Reducing the surface area of a container not only optimizes material usage but also improves energy efficiency, as less surface area means less heat loss or gain.