Introduction

When optimizing a rectangular prism, one common goal is to preserve the volume while varying the surface area. This article explores how to achieve this through mathematical manipulation and provides practical examples and formulas.

Step-by-Step Approach

Understanding the Formulas

To start, let's review the formulas for the volume and surface area of a rectangular prism.

Volume Formula

The volume V of a rectangular prism is given by the formula:

$$ V l times w times h $$

Where:

l is the length, w is the width, and h is the height.

Surface Area Formula

The surface area S is given by:

$$ S 2lw 2lh 2wh $$

Which simplifies to:

$$ S 2(lw lh wh) $$

Choosing Initial Dimensions

To find new dimensions that maintain the same volume but change the surface area, we can follow these steps:

Choose initial dimensions l1, w1, and h1 Calculate the volume:

$$ V l_1 times w_1 times h_1 $$

Select new dimensions l2, w2, and h2 such that V l2 times w2times h2 remains the same but S changes. You can adjust one or more dimensions to achieve this.

Example Calculation

Let’s use an example to illustrate these concepts. Suppose the initial dimensions are:

length (l1) 2 width (w1) 3 height (h1) 4

Calculate the volume:

$$ V 2 times 3 times 4 24 $$

Now, let’s try new dimensions. If we change the dimensions to:

l2 1 w2 6 h2 4

Calculate the new volume:

$$ V 1 times 6 times 4 24 $$

Now, calculate the surface areas:

Original Surface Area

$$ S_1 2(2 times 3) 2(2 times 4) 2(3 times 4) 12 16 24 52 $$

New Surface Area

$$ S_2 2(1 times 6) 2(1 times 4) 2(6 times 4) 12 8 48 68 $$

As expected, the volume remains 24, but the surface area has changed from 52 to 68.

Expressing Variables Algebraically

To further understand how to set the dimensions, we can express the variables algebraically:

$$ abc V $$

$$ ab frac{V}{c}, quad bc frac{V}{a}, quad ca frac{V}{b} $$

Substitute these into the surface area formula:

$$ S 2 left( frac{V}{c} right) 2 left( frac{V}{a} right) 2 left( frac{V}{b} right) 2V left( frac{1}{a} frac{1}{b} frac{1}{c} right) $$

NOW, abc are variables that you can set by choosing two of them and calculating the third.

Example Calculation Using Algebra

Let's use the algebraic approach with example dimensions:

a 3 b 4 c 5

Calculate the volume:

$$ V abc 3 times 4 times 5 60 $$

Calculate the surface area:

$$ S 2 left( frac{60}{5} right) 2 left( frac{60}{3} right) 2 left( frac{60}{4} right) 2 times 12 2 times 20 2 times 15 24 40 30 94 $$

Varying the Base

To further optimize the rectangular prism, you can consider varying the base while adjusting the height. One strategy is to double the base and reduce the height by half, as illustrated by the formula:

$$ V 2b times frac{1}{2}h $$

For example, if you double the base from 2 to 4, the height would need to be halved from 4 to 2:

$$ V 2 times 4 times 2 16 $$

Conclusion

By following these steps and formulas, you can easily manipulate the dimensions of a rectangular prism to achieve the same volume but a different surface area. This technique is valuable in a range of applications, from engineering to design.