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)

Question

Responses

1 inch
1 inch

4 inches
4 inches

9 inches
9 inches

3 inches

Answer 1

To find the width of the border Hassan wants to add, we need to first calculate the area of the picture itself and then relate it to the final area of the entire framed picture.

1. Calculate the area of the picture:

   • Length = 10 inches
   • Width = 6 inches
   • Area of the picture = Length × Width = 10 × 6 = 60 square inches

2. Determine the area that will be taken by the border:

   • Final area of the framed picture = 96 square inches
   • Therefore, the area of the border = Final area - Area of the picture = 96 - 60 = 36 square inches

3. Express the total dimensions of the framed picture: Let \( x \) be the width of the border. The dimensions of the framed picture will be:

   • Length = 10 + 2x (adding the border to both sides)
   • Width = 6 + 2x (adding the border to both sides)

4. Set up the equation using the total area: \[ (10 + 2x)(6 + 2x) = 96 \]

5. Expand the equation: \[ 60 + 20x + 12x + 4x^2 = 96 \] \[ 4x^2 + 32x + 60 = 96 \]

6. Rearranging the equation: \[ 4x^2 + 32x - 36 = 0 \]

7. Divide the entire equation by 4 to simplify: \[ x^2 + 8x - 9 = 0 \]

8. Factor the quadratic equation: \[ (x + 9)(x - 1) = 0 \]

9. Solve for \( x \):

   • \( x + 9 = 0 \) leads to \( x = -9 \) (not physically possible)
   • \( x - 1 = 0 \) leads to \( x = 1 \)

Thus, the width of the border is 1 inch.