Math Problem Statement

A rectangular prism with integer edge lengths is to be constructed so that its volume is within 10 cubic units of the maximum possible volume for a rectangular prism with edge lengths that are at most 5 units. The prism must also satisfy the condition that the sum of the lengths of any two of its edges is at least 6 units. What is the maximum possible volume of the prism?

Solution

To solve the problem systematically, let us proceed step by step.


1. Understanding the constraints

  • The edge lengths a,b,ca, b, c of the rectangular prism are integers, with each a,b,c5a, b, c \leq 5.
  • The volume V=abcV = a \cdot b \cdot c of the rectangular prism must be within 10 cubic units of the maximum possible volume for a rectangular prism with edge lengths at most 5.
  • The sum of the lengths of any two of its edges is at least 6 units.

Our goal is to find the maximum possible volume of the prism satisfying these conditions.


2. Maximum possible volume for a rectangular prism

If a,b,ca, b, c are all integers and at most 5, the maximum volume occurs when a=b=c=5a = b = c = 5. In this case:

Vmax=555=125.V_{\text{max}} = 5 \cdot 5 \cdot 5 = 125.

The condition states that the volume of the rectangular prism must be within 10 cubic units of this maximum. This means:

115V125.115 \leq V \leq 125.


3. Applying the second condition

The condition that the sum of the lengths of any two edges is at least 6 units imposes further restrictions:

  • For edge lengths a,b,ca, b, c, we require: a+b6,b+c6,andc+a6.a + b \geq 6, \quad b + c \geq 6, \quad \text{and} \quad c + a \geq 6.

This means that no edge length can be too small. For example:

  • If a=1a = 1, then bb and cc must be large enough to ensure b+c6b + c \geq 6.
  • Similarly for other edge combinations.

This constraint reduces the possible combinations for a,b,ca, b, c.


4. Searching for valid solutions

We now need to check all combinations of a,b,ca, b, c (where 1a,b,c51 \leq a, b, c \leq 5) to find those that:

  1. Have a volume V=abcV = a \cdot b \cdot c satisfying 115V125115 \leq V \leq 125,
  2. Satisfy a+b6a + b \geq 6, b+c6b + c \geq 6, and c+a6c + a \geq 6.

Step 4.1: Start with maximum volume V=125V = 125.

  • At a=5,b=5,c=5a = 5, b = 5, c = 5, the volume is 125125.
  • Check the edge sum conditions: a+b=5+5=10,b+c=5+5=10,c+a=5+5=10.a + b = 5 + 5 = 10, \quad b + c = 5 + 5 = 10, \quad c + a = 5 + 5 = 10. These clearly satisfy a+b6a + b \geq 6, b+c6b + c \geq 6, c+a6c + a \geq 6.

Thus, the prism with dimensions 5×5×55 \times 5 \times 5 satisfies all the conditions.


Step 4.2: Check other volumes VV within the range 115V<125115 \leq V < 125.

Next, we systematically test other combinations of a,b,ca, b, c to see if they satisfy the conditions. However:

  • Reducing any edge length from 5 will reduce the volume significantly.
  • For example:
    • If a=4,b=5,c=5a = 4, b = 5, c = 5, the volume is 455=1004 \cdot 5 \cdot 5 = 100, which is outside the required range.
    • If a=5,b=4,c=5a = 5, b = 4, c = 5, or similar combinations, the volume will again be less than 115.

Hence, no other combinations of a,b,ca, b, c with integer values can produce a volume in the range 115V125115 \leq V \leq 125 while satisfying the edge sum conditions.


5. Conclusion

The maximum possible volume of the prism that satisfies all the given conditions is:

125.\boxed{125}.

If you have further questions or need clarifications, feel free to ask!


Related Questions:

  1. What are the conditions for maximizing the volume of a rectangular prism?
  2. How does the sum condition a+b6a + b \geq 6 affect the possible edge lengths?
  3. What would happen if the edge lengths were not constrained to integers?
  4. How many possible combinations of edge lengths satisfy the conditions a5a \leq 5 and a+b6a + b \geq 6?
  5. Can a prism with a volume of 115 be constructed under the given conditions?

Tip:

When solving constrained optimization problems, systematically analyze the constraints and test boundary conditions to ensure all requirements are met.

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Math Problem Analysis

Mathematical Concepts

Geometry
Optimization
Inequalities

Formulas

Volume of a rectangular prism: V = a × b × c
Edge sum condition: a + b ≥ 6, b + c ≥ 6, c + a ≥ 6

Theorems

Optimization under constraints

Suitable Grade Level

Grades 8-10