Question
Nicky wants to surround a 25 in. by 16 in. painting with a frame that is w inches wide. Which quadratic function represents the combined area A of the picture and the frame
Answers
GPT-4o mini
To find the combined area \( A \) of the painting and the frame, we first need to express the dimensions of the framed painting in terms of \( w \), the width of the frame.
The original dimensions of the painting are:
- Width = 25 inches
- Height = 16 inches
When the frame is added, the width and height of the entire framed piece will increase by \( 2w \) (since the frame adds \( w \) to each side). Therefore, the new dimensions will be:
- Width of the framed painting = \( 25 + 2w \)
- Height of the framed painting = \( 16 + 2w \)
The area \( A \) of the combined painting and frame can be determined by multiplying the new width and height:
\[
A = (25 + 2w)(16 + 2w)
\]
Now we can expand this expression:
\[
A = 25 \times 16 + 25 \times 2w + 16 \times 2w + 2w \times 2w
\]
Calculating each term:
- \( 25 \times 16 = 400 \)
- \( 25 \times 2w = 50w \)
- \( 16 \times 2w = 32w \)
- \( 2w \times 2w = 4w^2 \)
Now combine all these terms:
\[
A = 400 + 50w + 32w + 4w^2
\]
\[
A = 400 + 82w + 4w^2
\]
Finally, we can write the quadratic function representing the combined area \( A \) as:
\[
A(w) = 4w^2 + 82w + 400
\]
So the quadratic function that represents the area of the picture and frame as a function of \( w \) is:
\[
A(w) = 4w^2 + 82w + 400
\]
The original dimensions of the painting are:
- Width = 25 inches
- Height = 16 inches
When the frame is added, the width and height of the entire framed piece will increase by \( 2w \) (since the frame adds \( w \) to each side). Therefore, the new dimensions will be:
- Width of the framed painting = \( 25 + 2w \)
- Height of the framed painting = \( 16 + 2w \)
The area \( A \) of the combined painting and frame can be determined by multiplying the new width and height:
\[
A = (25 + 2w)(16 + 2w)
\]
Now we can expand this expression:
\[
A = 25 \times 16 + 25 \times 2w + 16 \times 2w + 2w \times 2w
\]
Calculating each term:
- \( 25 \times 16 = 400 \)
- \( 25 \times 2w = 50w \)
- \( 16 \times 2w = 32w \)
- \( 2w \times 2w = 4w^2 \)
Now combine all these terms:
\[
A = 400 + 50w + 32w + 4w^2
\]
\[
A = 400 + 82w + 4w^2
\]
Finally, we can write the quadratic function representing the combined area \( A \) as:
\[
A(w) = 4w^2 + 82w + 400
\]
So the quadratic function that represents the area of the picture and frame as a function of \( w \) is:
\[
A(w) = 4w^2 + 82w + 400
\]