Question

A picture measuring 4 inches by 6 inches is placed inside a frame which has equal width around the entire picture. If the area of the frame and the picture is 63 square inches, what is the width of the frame

Answer 1

1.5 inches.

Explanation:

A picture measuring 4 inches by 6 inches is placed inside a frame which has equal width around the entire picture.

The area of the frame and the picture is 63 square inches.

Question asked:

What is the width of the frame ?

Solution:

Let width of the frame =
`x`

Then combined length (frame and picture) =
`6+x+x=6+2x\ inches`

Similarly, combined width (frame and picture) =
`4+x+x=4+2x \ inches`

By using:-
`Area\ of\ rectangle=length* width`

Combined area of the frame and the picture = 63 square inches

-
`length* width=63\\(6+2x)(4+2x)=63\\\\6(4+2x)+2x(4+2x)=63\\24+12x+8x+4x^(2) =63\\24+20x+4x^(2) =63`

Subtracting both sides by 24

`4x^(2) +20x= 39`

Subtracting both sides by 39

`4x^(2) +20x-39= 39-39\\4x^(2) +20x-39=0`

`4x^(2) + 26x - 6x-39 = 0\\Taking\ common\\`

`2x(2x+13)-3 (2x+13)=0\\2x+13=0,2x-3=0\\2x=0-13,2x= 0+3`

`x=(-13)/(2) ,x=(3)/(2)`

Since, width can never be in negative thus,
`x=(3)/(2) = 1.5 \ inches`

Therefore, width of the frame is 1.5 inches around the entire picture.