A rectangular floor measures $a$ feet by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a r

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A rectangular floor measures $a$ feet by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a r

ectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 foot around the painted rectangle and occupies half the area of the entire floor. How many possibilities are there for the ordered pair $(a,b)$?

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1 answer:

DENIUS [597] · Answer 1 · 2023-04-14T11:20:08+03:00

The number of possibilities that are there for the ordered pair is 2.

<h3>Calculation and Parameters
</h3>

Because the unpainted part of the floor covers 1/2 of the area, then the painted rectangle covers 1/2 of the area as well.

Given that the border width is 1 foot, the dimensions of the rectangle are a-2 by b-2.

Making an equation:

ab= 2((a-2)(b-2))

ab= 2ab-4a-4b+8

ab-4a-4b+8=0

If we apply Simon's Favorite Factoring Trick,

ab-4a-4b+16= 8

(a-4)(b-4)= 8

Since b > a, then we have the possibilities

(a-4) = 1 and (b-4) = 8, or (a-4) = 2 and (b-4) = 4.

(5, 12) or (6,8) which gives us 2 possibilities.