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?

Answer 1

To find the width of the border that Hassan wants to add to the photo, we can start by calculating the area of the picture itself, then determine the total area of the framed picture, and finally find the dimensions of the framed picture to calculate the width of the border.

1. **Calculate the area of the picture:**
\[
\text{Area of the picture} = \text{length} \times \text{width} = 10 \, \text{inches} \times 6 \, \text{inches} = 60 \, \text{square inches}
\]

2. **Determine the area of the framed picture:**
The final area of the entire framed picture is given as 96 square inches.

3. **Set up the equation for the area of the framed picture:**
Let \( x \) be the width of the border. The dimensions of the framed picture will then be:
- Length: \( 10 + 2x \)
- Width: \( 6 + 2x \)

The area of the framed picture can be expressed as:
\[
(10 + 2x)(6 + 2x) = 96
\]

4. **Expand and simplify the equation:**
\[
(10 + 2x)(6 + 2x) = 60 + 20x + 12x + 4x^2 = 60 + 32x + 4x^2
\]

Setting this equal to the total area:
\[
4x^2 + 32x + 60 = 96
\]

Subtracting 96 from both sides gives:
\[
4x^2 + 32x + 60 - 96 = 0
\]
\[
4x^2 + 32x - 36 = 0
\]

5. **Simplify the quadratic equation:**
We can divide the entire equation by 4:
\[
x^2 + 8x - 9 = 0
\]

6. **Factor the quadratic equation:**
To factor \( x^2 + 8x - 9 = 0 \), we look for two numbers that multiply to \(-9\) and add to \(8\). The factors are \(9\) and \(-1\):
\[
(x + 9)(x - 1) = 0
\]

7. **Solve for \( x \):**
Setting each factor to zero gives:
\[
x + 9 = 0 \quad \Rightarrow \quad x = -9 \quad (\text{not valid since width cannot be negative})
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

Thus, the width of the border is \( \boxed{1} \) inch.