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. What is the width of the border?

Answer 1

To find the width of the border that Hassan wants to add to his photo, we can start by calculating the area of the frame including the border and subtracting the area of the photo itself.

The area of the entire framed picture is given as 96 square inches. The dimensions of the photo are:
- Length \( L = 10 \) inches
- Width \( W = 6 \) inches

First, we calculate the area of the photo:

\[
\text{Area of the photo} = L \times W = 10 \times 6 = 60 \text{ square inches}
\]

Next, we know the area of the framed picture, which is the total area including the border:

\[
\text{Area of the framed picture} = 96 \text{ square inches}
\]

Now we can find the area of the border by subtracting the area of the photo from the area of the framed picture:

\[
\text{Area of the border} = \text{Area of the framed picture} - \text{Area of the photo} = 96 - 60 = 36 \text{ square inches}
\]

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

Now we can write an equation based on the area of the framed picture:

\[
(10 + 2x)(6 + 2x) = 96
\]

Expanding this equation:

\[
60 + 20x + 12x + 4x^2 = 96
\]

Combining the like terms gives us:

\[
4x^2 + 32x + 60 = 96
\]

Now, we can simplify this equation by subtracting 96 from both sides:

\[
4x^2 + 32x + 60 - 96 = 0
\]
\[
4x^2 + 32x - 36 = 0
\]

Next, we can divide the entire equation by 4 to simplify:

\[
x^2 + 8x - 9 = 0
\]

Now, we will factor this quadratic equation. We need two numbers that multiply to \(-9\) and add up to \(8\). These numbers are \(9\) and \(-1\):

\[
(x + 9)(x - 1) = 0
\]

Setting each factor to zero gives us:

\[
x + 9 = 0 \quad \Rightarrow \quad x = -9 \quad \text{(not a valid solution)}
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

Since negative width doesn't make sense in this context, we have:

\[
x = 1
\]

Thus, the width of the border is

\[
\boxed{1} \text{ inch}.
\]