GPT答：

7i24.cn

Let x be the width of the printed area and y be the height of the printed area. Then we have:

xy = 81 (since the printed area has an area of 81 cm2)

The total width of the page is x + 4 (2 cm on each side), and the total height of the page is y + 6 (3 cm at top and bottom). The total area of paper used is then:

A = (x + 4)(y + 6)

We want to minimize A subject to xy = 81. Using substitution, we can eliminate y from this equation:

y = 81/x

Substituting into A, we get:

A(x) = (x+4)(81/x+6)

Expanding and simplifying, we get:

A(x) = x^2/9 + 18x +216/x +864/9

To find where A(x) has a minimum, we take its derivative with respect to x and set it equal to zero:

A'(x) = -x^2/81 -216/x^2 +18

0= -x^4/6561-216+x^2*1458
0=x^6-1458*x^2-1417176
Using a calculator or numerical methods, we find that this equation has one positive root: approximately x ≈14.7 cm.

Substituting back into xy=81 gives us y≈5.51cm.

Therefore, the dimensions that minimize paper usage are approximately 14.7 cm by 5.51 cm.

To show that this point is a minimum rather than a maximum or inflection point, we can check that A''(14.7)>0 which means it's concave up at that point so it must be a minimum value for our function A(x).