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)

Responses

1 inch
1 inch

9 inches
9 inches

3 inches
3 inches

4 inches

1 answer

To find the width of the border, let's denote the width of the border as \( x \) inches. The dimensions of the framed photo will then be:

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

The area of the framed picture can be expressed as:

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

Expanding this equation:

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

Combining like terms gives:

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

Subtracting 96 from both sides:

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

Dividing the entire equation by 4 to simplify:

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

Now we can factor the quadratic equation:

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

Setting each factor equal to zero gives us:

\[ x + 9 = 0 \quad \Rightarrow \quad x = -9 \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

Since the width of the border cannot be negative, we have:

\[ x = 1 \]

So the width of the border is 1 inch.