Problem 37 - A History of Mathematics: An Introduction

Question

Divide 8 into two parts \(x, y\), such that \(x y(x-y)\) is a maximum. (Note that this was posed in the days before calculus.)

Step-by-Step Solution

Answer

Question: Using algebraic manipulation, find the two parts x and y that divide the number 8 so that their product x*y(x-y) is maximized. Answer: The maximum value of the product occurs when x = 4 and y = 4, and the product is 32.

Step 1: Set up the problem

Since we want to divide the number 8 into two parts \(x\) and \(y\), we must have \(x+ y = 8\). We can rearrange this equation to solve for either \(x\) or \(y\). For example, we can find \(x\) as \(x= 8-y\).

Step 2: Substitute x in the product equation

We now want to maximize the product \(x y(x-y)\). We can substitute the expression we found for \(x\) in step 1 into this equation: \((8-y) y((8-y)-y)\).

Step 3: Simplify the expression

We can now simplify the expression and find the maximum value of the product: \((8-y) y(8-2y) = (8y-y^2)(8-2y)\).

Step 4: Factor the quadratic

Expand the expression to have a quadratic in terms of \(y\): \((8y-y^2)(8-2y) = -2y^3+16y^2-64y\).

Step 5: Identify the critical points of the cubic

Now, we have a cubic function, and we want to find the maximum value of the function (-2y^3+16y^2-64y). A crude approach will be to observe the cubic polynomial, given that \(0 \leq y \leq 8\), as x and y are parts of 8.

Step 6: Test the polynomial at the bounds

Let's evaluate the polynomial at y=0 and y=8: At \(y=0\), the function value is \(-2(0)^3 + 16(0)^2 - 64(0) = 0\). At \(y=8\), the function value is \(-2(8)^3 + 16(8)^2 - 64(8) = 0\).

Step 7: Analyze the behavior of the cubic

Given that the function is zero at the boundaries, we can conclude that there must be some point within the bounds of the function where it achieves a maximum value. We observe that the function is symmetric around the line \(y=4\), because it consists of even and odd-degree terms.

Step 8: Find x and y values for maximum product

since the function is symmetric around the line \(y=4\), we can plug \(y=4\) back into the equation from step 1 to find \(x\): \(x=8-y= 8-(4)=4\). The maximum value of the product \(x y(x-y)\) occurs when both \(x\) and \(y\) are equal to \(4\), and the product is \(4(4)(4-4)= 32\).