Question

Hassan wants to add a border to a photo he took before he frames it. The final area of the entire framed picture will be 96 square inches. The length of the picture is 10 inches, and the width is 6 inches. Which of the following is the width of the border? (1 point)

1 inch

3 inches

9 inches

4 inches

Answer 1

To find the width of the border, we first need to determine the area of the actual picture and then the area of the entire framed picture.

The dimensions of the picture are:
- Length = 10 inches
- Width = 6 inches

The area of the picture is calculated as follows:
\[
\text{Area of the picture} = \text{Length} \times \text{Width} = 10 \, \text{inches} \times 6 \, \text{inches} = 60 \, \text{square inches}
\]

Next, we know that the final area of the entire framed picture is:
\[
\text{Area of the framed picture} = 96 \, \text{square inches}
\]

The area of the border around the picture can be found by subtracting the area of the picture from the area of the framed picture:
\[
\text{Area of the border} = \text{Area of the framed picture} - \text{Area of the picture} = 96 \, \text{square inches} - 60 \, \text{square inches} = 36 \, \text{square inches}
\]

Let \( x \) be the width of the border. The dimensions of the framed picture (including the border) will be:
- Length = \( 10 + 2x \)
- Width = \( 6 + 2x \)

The area of the framed picture in terms of \( x \) is:
\[
\text{Area of the framed picture} = (10 + 2x)(6 + 2x)
\]

Setting this equal to the area of the framed picture, we have:
\[
(10 + 2x)(6 + 2x) = 96
\]

Expanding the left side:
\[
60 + 20x + 12x + 4x^2 = 96
\]
\[
60 + 32x + 4x^2 = 96
\]

Now, subtract 96 from both sides:
\[
4x^2 + 32x + 60 - 96 = 0
\]
\[
4x^2 + 32x - 36 = 0
\]

Now, we can simplify by dividing all terms by 4:
\[
x^2 + 8x - 9 = 0
\]

Next, we can factor this quadratic equation:
\[
(x + 9)(x - 1) = 0
\]

Setting each factor equal to zero gives us:
\[
x + 9 = 0 \quad \Rightarrow \quad x = -9 \quad \text{(not valid since width can't be negative)}
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

Hence, the width of the border is:
\[
\boxed{1} \, \text{inch}
\]