Maximizing the Value of (a^2b) Given (ab 20)
Consider the problem of finding the maximum value of (a^2b) given the constraint (ab 20), where (a) and (b) are positive integers.
Using Calculus for Optimization
Given:
[ab 20]
From this, we can express (b) in terms of (a):
[b frac{20}{a}]
Let (f(a) a^2 cdot b). Substituting the expression for (b), we get:
[f(a) a^2 cdot frac{20}{a} 20a]
However, the correct expression should be:
[f(a) a^2 cdot frac{20}{a} 20a - a^2]
To find the maximum value, we take the derivative of (f(a)) with respect to (a):
[frac{df}{da} 40a - 3a^2]
Setting the derivative to zero for critical points:
[40a - 3a^2 0]
[a(40 - 3a) 0]
This gives us:
[a 0 quad text{or} quad 40 - 3a 0]
[a frac{40}{3}]
Since (a frac{40}{3}) is not an integer, we cannot use this value. Therefore, we need to consider the nearest integers to (frac{40}{3}), which are (a 13) and (a 14).
Checking (f(a)) for these values:
For (a 13):
[b frac{20}{13} approx 1.538]
(Not an integer, so discard this option)
For (a 14):
[b frac{20}{14} approx 1.429]
(Not an integer, so discard this option)
The nearest integer pair that satisfies (ab 20) is (a 13) and (b 1.538), approximately, which can be approximated as (a 13) and (b 7) for integer solutions.
For (a 13):
[b frac{20}{13} approx 1.538]
(Not an integer, so discard this option)
For (a 7) and (b frac{20}{7} approx 2.857), approximately:
[a^2b 7^2 cdot frac{20}{7} 49 cdot frac{20}{7} 7 cdot 20 140]
Using Arithmetic and Geometric Mean Inequality
Another method to find the maximum value is by using the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
[frac{frac{a}{2} frac{a}{2} b}{3} geq sqrt[3]{frac{a}{2} cdot frac{a}{2} cdot b}]
Simplifying, we get:
[frac{a b}{3} geq sqrt[3]{frac{a^2b}{4}}]
[left(frac{a b}{3}right)^3 geq frac{a^2b}{4}]
[frac{a^2b}{4} leq frac{20^3}{3^3}]
[frac{a^2b}{4} leq frac{8000}{27}]
[a^2b leq frac{32000}{27}]
The integer part of (frac{32000}{27}) is 1185. So, a possible maximum value of (a^2b) could be 1185.
Conclusion: The nearest integer solution yielding the maximum value for (a^2b) given (ab 20) is (a 13) and (b 7), providing:
[boxed{13^2 cdot 7 1183}]