According to the conditions in the problem, the equation can be obtained.

(4-2 times) +(2πx÷2)？ ≤4? , and

(4-2 times) +(2πx)？ ≥4? ;

Have 16- 16x+4x? +π? x？ ≤ 16,

4-4x+x？ +π? x？ ≥4;

(4+π? )x？ - 16x≤0，( 1+π？ )x？ -4x≥0, so:

4/( 1+π? )≤x≤ 16/(4+π？ )

The surface area of tin bucket S=2πx? +2πx×(4×2-4x)，

S=2πx？ + 16πx-8πx？ ，S= 16πx-6πx？ ,

S=-6π(x？ -8x/3+ 16/9)+32π/3，

S=-6π(x-4/3)？ +32π/3

∵4/3> 16/(4+π? ) ∴ When x= 16/(4+π? ),

maxS=256π/(4+π？ )- 1536π/(4+π? )? ,

maxS=256π(4+π？ -6π)/(4+π? )

Please refer to

Equations with unknowns are equations, and mathematics was first developed in counting. The combination of addition, subtraction, multiplication, division and idempotency between numbers and unknowns forms algebraic equations: one-dimensional linear equations, one-dimensional quadratic equations, two-dimensional linear equations and so on. However, with the emergence of the concept of function and the introduction of differential and integral operation based on function, the category of equation is wider, and the unknowns can be mathematical objects such as functions and vectors, and the operation is no longer limited to addition, subtraction, multiplication and division.

Equation occupies an important position in mathematics and seems to be an eternal topic in mathematics. The appearance of equations not only greatly expanded the application scope of mathematics, but also solved many problems that could not be solved by arithmetic problem-solving methods, which had a great influence on the progress of mathematics later. In particular, many important discoveries in mathematics are closely related to it.

A system of equations with two binary variables and two squares

Because mathematics has changed from constant mathematics to variable mathematics, the contents of equations are enriched, because mathematics has introduced more concepts and more operations, thus forming more equations. The development of other natural sciences, especially physics, also directly puts forward the demand of solving equations and provides a large number of research topics.