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Answer 1

The dimensions based on the area given are 15cm by 20cm.

How to calculate the sides?

From the information given about the rectangle, the area is given as 300cm² and the margins are given as 1.5 and 2.

Therefore, the dimensions will be:

(1.5 × x) × (2 × x) = 300

1.5x × 2x = 300

3x² = 300

x² = 300/3

x² = 100

x = 10

Therefore the dimensions will be:

= 1.5x = 1.5 × 10 = 15

= 2x = 2 × 10 = 20

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Answer 2

We minimize the total area of the sheet

$S = xy$

constrained by

$(x-4)(y-3) = 300$

Solve the constraint equation for $y$.

$(x-4)(y-3) = 300 \implies y-3 = \dfrac{300}{x-4} \implies y = 3 + \dfrac{300}{x-4} = \dfrac{3x+288}{x-4}$

Substitute this into $S$ and find the critical points.

$S = \dfrac{3x^2+288x}{x-4} = 3x + 300 + \dfrac{1200}{x-4}$

$\dfrac{dS}{dx} = 3 - \dfrac{1200}{(x-4)^2} = 0$

$\implies \dfrac{1200}{(x-4)^2} = 3$

$\implies (x-4)^2 = 400$

$\implies x-4 = \pm20$

$\implies x=-16 \text{ or } x = 24$

Of course $x$ can't be negative, so the page dimensions that minimize $S$ are $x=24$ and $y=18$.