To find the width of the border, we first need to calculate the area of the picture without the border. The area of the picture is given by its dimensions:
\[ \text{Area of the picture} = \text{length} \times \text{width} = 10 , \text{inches} \times 6 , \text{inches} = 60 , \text{square inches} \]
Next, we know the total area of the framed picture is 96 square inches. The border area can be calculated by subtracting the area of the picture from the total area:
\[ \text{Area of the border} = \text{Total area} - \text{Area of the picture} = 96 , \text{square inches} - 60 , \text{square inches} = 36 , \text{square inches} \]
Let \( w \) be the width of the border. The dimensions of the entire framed picture will be:
\[ \text{Length of framed picture} = 10 + 2w \] \[ \text{Width of framed picture} = 6 + 2w \]
The area of the framed picture can also be expressed as:
\[ \text{Area of framed picture} = (10 + 2w)(6 + 2w) \]
Setting this equal to the total area:
\[ (10 + 2w)(6 + 2w) = 96 \]
Expanding the left side:
\[ 60 + 20w + 12w + 4w^2 = 96 \] \[ 4w^2 + 32w + 60 = 96 \]
Now, simplifying:
\[ 4w^2 + 32w + 60 - 96 = 0 \] \[ 4w^2 + 32w - 36 = 0 \]
Dividing the entire equation by 4:
\[ w^2 + 8w - 9 = 0 \]
Next, we can factor the quadratic:
\[ (w + 9)(w - 1) = 0 \]
Setting each factor to zero gives us:
\[ w + 9 = 0 \quad \Rightarrow \quad w = -9 \quad \text{(not valid, as width cannot be negative)} \] \[ w - 1 = 0 \quad \Rightarrow \quad w = 1 \]
So, the width of the border is 1 inch.
Thus, the correct response is:
1 inch
1 answer