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?
A) 1 inch
B) 9 inches
C) 4 inches
D) 3 inches

Answer 1

Let's denote the width of the border as \( x \).

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

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

We know that the total area of the framed picture is 96 square inches. Thus, we can set up the following equation:
\[
(10 + 2x)(6 + 2x) = 96
\]

Now, we will expand the left side:
\[
10 \cdot 6 + 10 \cdot 2x + 6 \cdot 2x + 2x \cdot 2x = 60 + 20x + 12x + 4x^2
\]
\[
= 60 + 32x + 4x^2
\]

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

We will move 96 to the left side of the equation:
\[
4x^2 + 32x + 60 - 96 = 0
\]
\[
4x^2 + 32x - 36 = 0
\]

Next, we can simplify the equation by dividing everything by 4:
\[
x^2 + 8x - 9 = 0
\]

Now we will factor this quadratic equation. We need two numbers that multiply to \(-9\) and add to \(8\). The numbers \(9\) and \(-1\) satisfy this condition:
\[
(x + 9)(x - 1) = 0
\]

Setting each factor to zero gives us:
\[
x + 9 = 0 \quad \Rightarrow \quad x = -9 \quad (\text{not a valid width})
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

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