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)

Answer 1

To find the width of the border that Hassan wants to add, let's denote the width of the border as \( x \).

The dimensions of the framed picture, including the border, will be:

- Length: \( 10 + 2x \) (adding the border to both sides)
- Width: \( 6 + 2x \) (adding the border to both sides)

The area of the entire framed picture is given as 96 square inches. Therefore, we can set up the following equation for the area:

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

Expanding the left side of the equation:

\[
10 \times 6 + 10 \times 2x + 6 \times 2x + 4x^2 = 96
\]

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

Combining like terms:

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

Next, we can set the equation to zero by subtracting 96 from both sides:

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

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

Now, we can simplify this equation by dividing the entire equation by 4:

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

Next, we can factor the quadratic equation:

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

Setting each factor to zero gives us:

\[
x + 9 = 0 \quad \Rightarrow \quad x = -9 \quad (\text{not a valid solution since width cannot be negative})
\]

\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

Thus, the width of the border \( x \) is \( 1 \) inch.

So, the width of the border is **1 inch**.