Question

A square picture is to be mounted on a frame with a border of width 1 centimetre.if the area of the border is three times, that area of the picture, find the length and side of the picture

Answer 1

Let side length of square picture is P cm
then,
area of square picture = (side length) ²
= (P)² = P² cm²

border width = 1 cm
so, outer side length of frame = (P+2) cm
inner side length of frame = P cm
now,
area of border = area of outer square - area of inner square
= (P+2)² - P²
= P² + 4P +4 - P²
=( 4P + 4 ) cm

now, a/c to question,
3 × area of border = area of picture
3 (4P + 4) = P²
12P + 12= P²
P² -12P -12 = 0
use quadratic equation,
P = {12 ±√(12² + 4×12)}/2
= 6 ± 4√3
P ≠ 6 - 4√3 [ length doesn't negative ]

so, P = (6+√3) cm
hence, side length of picture = (6+√3) cm