Question

6 A 4-inch by 6-inch photo is put into a picture frame with a border ofconstant width. If the area of the frame, including the picture, is 80 squareinches, find an equation for determining the width of the border and use thatequation to solve for it. Show all work.

Answer 1

Assume that the width of the frame is x, then

The length and the width of the picture with the frame will increase by 2x (x for each side), then

L = 6 + 2x

W = 4 + 2x

Since the total area is 80 in.^2, then

`L* W=80`

Substitute the values of L and W

`(6+2x)(4+2x)=80`

Now, we will solve the equation

`\begin{gathered} (6+2x)(4+2x)=(6)(4)+(6)(2x)+(4)(2x)+(2x)(2x) \\ (6+2x)(4+2x)=24+12x+8x+4x^2 \end{gathered}`

Add the like terms

`(6+2x)(4+2x)=24+20x+4x^2`

Now, equate it by 80

`4x^2+20x+24=80`

Subtract 80 from both sides

`\begin{gathered} 4x^2+20x+24-80=80-80 \\ 4x^2+20x-56=0 \end{gathered}`

Divide all terms by 4 to simplify the equation

`\begin{gathered} (4x^2)/(4)+(20x)/(4)-(56)/(4)=(0)/(4) \\ x^2+5x-14=0 \end{gathered}`

The equation is

`x^2+5x-14=0`

Now, factorize it into 2 factors

`\begin{gathered} x^2=(x)(x) \\ -14=(7)(-2) \\ 7x-2x=5x \\ (x+7)(x-2)=0 \end{gathered}`

Equate each factor by 0 to find x

`\begin{gathered} x+7=0 \\ x+7-7=0-7 \\ x=-7 \\ x-2=0 \\ x-2+2=0+2 \\ x=2 \end{gathered}`

We will refuse x = -7 because the length must be a positive number, then

The width of the frame is 2 inches