Question

a rectangular picture is 6 centimeters longer than it is wide. A frame 1 cm wide is placed around the picture. Write a polynomial in simplest form that represents the perimeter of the frame. Then write a polynomial in simplest form that represents the area of the frame, not including the picture. Lastly, find the perimeter of the frame if the picture is 15 centimeters wide.

Answer 1

To find:-

• To write a polynomial representing the perimeter of the frame .
• Area of the frame , excluding the picture.
• Perimeter of the frame if picture is 15cm wide .

Answer:-

We are here given that a frame of width 1cm is placed around the picture whose length is 6cm more than its breadth. So let's take the breadth be " x " , then its length will be, " x + 6 "

Answer 1 :-

For diagram refer to the attachment.

Length of the frame = AB = AI + IJ + JB

= 1 + x + 6 + 1

= ( x + 8 ) cm

Breadth of the frame = BC = JL + LG + GK

= 1 + x + 1

= ( x + 2 ) cm

So the perimeter can be calculated using the formula,

P = 2 ( l + b )

P = 2 ( x + 8 + x + 2) cm

P = 2 ( 2x + 10 ) cm

P = ( 4x + 20 ) cm

Hence the perimeter of the frame is ( 4x + 20 ) cm .

_________________________________________

Answer 2 :-

For finding the area of the frame , subtract the area of picture from the total area of picture and frame . Let the required area be " S " , then ;

S = ∆A

S = (x+8)(x+2) - x(x+6)

S = x² + 2x + 8x + 16 - ( x² + 6x )

S = x² + 10x + 16 - x² - 6x

S = ( 4x + 16 ) cm²

Hence the area of the frame is ( 4x + 16 ) cm² .

_________________________________________

Answer 3 :-

In answer 1 we found the perimeter of the frame to be ( 4x + 20 ) cm . So if width of the picture that is " x " is 15cm , then the perimeter can be calculated by substituting x = 15 , in the obtained expression as ,

P = ( 4*15 + 20 ) cm

P = ( 60 + 20 ) cm

P = 80 cm

Hence the perimeter of the frame is 80cm .

and we are done!

Answer 2

`\textsf{Perimeter of frame:} \quad P(x)=4x+20`

`\textsf{Area of frame:} \quad A(x)=4x+16`

The perimeter of the frame is 80 cm.

Explanation:

What is a rectangle?

A rectangle is a two-dimensional shape where all the interior angles are right angles (90°) and its opposite sides are equal in length and parallel.

What is a perimeter?

The perimeter of a two-dimensional shape is the distance all the way around the outside.

Given a rectangular picture is 6 cm longer than it is wide:

• width = x cm
• length = (x + 6) cm

If a frame 1 cm wide is placed around the picture, then the width and length will be extended by 2 cm. Therefore:

• width = (x + 2) cm
• length = (x + 8) cm

Since the perimeter of a rectangle is twice the sum of its width and length, the polynomial that represents the perimeter of the frame is:

`\begin{aligned}\implies P(x)&=2(\sf width+length)\\&=2(x+2+x+8)\\&=2(2x+10)\\&=4x+20\end{aligned}`

The area of a rectangle is the product of its length and width.

The area of the frame is the area of the largest rectangle less the area of the picture. Therefore, the polynomial that represents this is:

`\begin{aligned}\implies A(x)&=(x+2)(x+8)-x(x+6)\\&=x^2+10x+16-x^2-6x\\&=x^2-x^2+10x-6x+16\\&=4x+16\end{aligned}`

To find the perimeter of the frame if the picture is 15 cm wide, substitute x = 15 into the polynomial P(x):

`\begin{aligned}x=15\implies P(15)&=4(15)+20\\&=60+20\\&=80\; \sf cm\end{aligned}`

Therefore, the perimeter of the frame is 80 cm.