Respuesta :
Answer:
Part a) The polynomial for the area including the frame is
[tex](4x^{2}+72x+320)\ ft^{2}[/tex]
Part b) The polynomial for the perimeter including the frame is
[tex](72+8x)\ ft[/tex]
Part c) The perimeter of the picture including the frame when the width of the frame is 2 inches is equal to [tex]88\ ft[/tex]
Part d) The area of the picture including the frame when the width of the frame is 2 inches is equal to [tex]480\ ft^{2}[/tex]
Step-by-step explanation:
Let
x-----> the width of the frame
L-----> the length of the picture
W-----> the width of the picture
Part a) What is a polynomial for the area including the frame?
we have
The dimensions of the picture are
[tex]L=20\ in[/tex]
[tex]W=16\ in[/tex]
The area including the frame is equal to
[tex]A=(20+2x)(16+2x)\\ \\A=320+40x+32x+4x^{2}\\ \\A=(4x^{2}+72x+320)\ ft^{2}[/tex]
Part b) What is a polynomial for the perimeter including the frame?
we have
The dimensions of the picture are
[tex]L=20\ in[/tex]
[tex]W=16\ in[/tex]
The perimeter including the frame is equal to
[tex]P=2[(20+2x)+(16+2x)]\\ \\P=2[36+4x]\\ \\P=(72+8x)\ ft[/tex]
Part c) What is the perimeter of the picture including the frame when the width of the frame is 2 inches
we have
[tex]P=(72+8x)\ ft[/tex]
For x=2 in
substitute
[tex]P=72+8(2)=88\ ft[/tex]
Part d) What is the area of the picture including the frame when the width of the frame is 2 inches
we have
[tex]A=(4x^{2}+72x+320)\ ft^{2}[/tex]
For x=2 in
substitute
[tex]A=(4(2)^{2}+72(2)+320)=480\ ft^{2}[/tex]
The perimeter of a two-dimensional figure is the distance covered around it.
The polynomial for the area including the frame is [tex]\rm 4x^2+72x +320[/tex].
The polynomial for the perimeter including frame is [tex]\rm 72+8x[/tex].
The perimeter of the picture including the frame when the width of the frame is 2 inches is 88 feet.
The area of the picture including the frame when the width of the frame is 2 inches is 480 feet.
Given that
You are framing a picture with a frame of equal width on each side.
The width of the frame is 2 inches.
The length is 20 inches and the width is 16 inches.
The width of the frame is 2 inches.
What is the perimeter?
The perimeter of a two-dimensional figure is the distance covered around it.
Let the width of the frame be x.
The length of the picture be L.
The width of the picture is W.
1. What is a polynomial for the area including the frame?
[tex]\rm Area \ of \ the \ frame = length \times width\\\\ Area \ of \ the \ frame = (20+2x) (16+2x)\\\\ Area \ of \ the \ frame = 320+40x+32x+4x^2\\\\ Area \ of \ the \ frame = 4x^2+72x +320[/tex]
The polynomial for the area including the frame is [tex]\rm 4x^2+72x +320[/tex].
2. What is a polynomial for the perimeter including the frame?
[tex]\rm Perimeter \ of \ the \ picture = 2 (length + width)\\\\ Perimeter \ of \ the \ picture = 2(20+2x+16+2x)\\\\ Perimeter \ of \ the \ picture = 2(36+4x)\\\\ Perimeter \ of \ the \ picture = 72+8x[/tex]
The polynomial for the perimeter including frame is [tex]\rm 72+8x[/tex].
3. What is the perimeter of the picture including the frame when the width of the frame is 2 inches.
[tex]\rm Perimeter \ of \ the \ picture = 72+8x\\\\Perimeter \ of \ the \ picture = 72+8(2)\\\\ Perimeter \ of \ the \ picture = 72+16\\\\Perimeter \ of \ the \ picture = 88\\\\[/tex]
The perimeter of the picture including the frame when the width of the frame is 2 inches is 88 feet.
4. What is the area of the picture including the frame when the width of the frame is 2 inches.
[tex]\rm Area \ of \ the \ frame = 4x^2+72x +320\\\\ Area \ of \ the \ frame = 4(2)^2+72(2)+320\\\\ Area \ of \ the \ frame=480[/tex]
The area of the picture including the frame when the width of the frame is 2 inches is 480 feet.
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