9. (Use the diagram to answer questions: A picture is $$ 20 \mathrm{~cm} imes 30 \mathrm{~cm} $$. a) the frame is $$ x \mathrm{~cm} $$, find an expression for the area of the picture And frame: $$ (A=l imes w) $$

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9. (Use the diagram to answer questions: A picture is $$ 20 \mathrm{~cm} 	imes 30 \mathrm{~cm} $$. a) the frame is $$ x \mathrm{~cm} $$, find an expression for the area of the picture And frame: $$ (A=l 	imes w) $$

UpStudy AI · Accepted Answer

We begin by noting that the picture has dimensions $$20 \, \text{cm}$$ by $$30 \, \text{cm}$$. When a frame of width $$x \, \text{cm}$$ is added all around the picture, the overall dimensions increase by $$2x$$ in each direction (since the frame appears on both sides).

1. The new width becomes:
   $$
   20 + 2x \, \text{cm}
   $$
2. The new length becomes:
   $$
   30 + 2x \, \text{cm}
   $$

The area $$A$$ of the picture and frame together is given by multiplying the new dimensions:
$$
A = (20+2x)(30+2x)
$$

3. Expanding the expression:
$$
A = 20 \times 30 + 20 \times 2x + 2x \times 30 + 2x \times 2x
$$
$$
A = 600 + 40x + 60x + 4x^2
$$

4. Combine like terms:
$$
A = 4x^2 + 100x + 600
$$

Thus, the expression for the area of the picture and frame is:
$$
\boxed{4x^2 + 100x + 600}
$$
The area of the picture and frame is $$4x^2 + 100x + 600$$.

9. (Use the diagram to answer questions: A picture is \( 20 \mathrm{~cm} \times 30 \mathrm{~cm} \). a) the frame is \( x \mathrm{~cm} \), find an expression for the area of the picture And frame: \( (A=l \times w) \)

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