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)Responses9 inches9 inches3 inches3 inches4 inches4 inches1 inch

Answer 1

To solve the problem, let's denote the width of the border as \( b \).

The dimensions of the photo itself are:
- Length: 10 inches
- Width: 6 inches
- Area of the photo = Length × Width = \( 10 \times 6 = 60 \) square inches.

When we add a border of width \( b \) around the photo, the new dimensions of the framed picture become:
- Length of the framed picture = \( 10 + 2b \)
- Width of the framed picture = \( 6 + 2b \)

The area of the entire framed picture is given as 96 square inches, so we set up the equation for the area:
\[
(10 + 2b)(6 + 2b) = 96
\]

Now, expanding the left side:
\[
10 \times 6 + 10 \times 2b + 6 \times 2b + 2b \times 2b = 60 + 20b + 12b + 4b^2 = 60 + 32b + 4b^2
\]

Setting the equation:
\[
60 + 32b + 4b^2 = 96
\]

Subtracting 96 from both sides:
\[
4b^2 + 32b + 60 - 96 = 0 \quad \Rightarrow \quad 4b^2 + 32b - 36 = 0
\]

Dividing the entire equation by 4 to simplify:
\[
b^2 + 8b - 9 = 0
\]

Now we can factor this quadratic:
\[
(b + 9)(b - 1) = 0
\]

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

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