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)
Responses
4 inches
4 inches
9 inches
9 inches
1 inch
1 inch
3 inches
Responses
4 inches
4 inches
9 inches
9 inches
1 inch
1 inch
3 inches
Answers
GPT-4o mini
To find the width of the border, we need to determine the dimensions of the framed picture and set up an equation for the area.
1. The area of the picture itself is calculated as follows:
\[
\text{Area of the picture} = \text{length} \times \text{width} = 10 \, \text{inches} \times 6 \, \text{inches} = 60 \, \text{square inches}
\]
2. The area of the entire framed picture is given as 96 square inches.
3. The area of the border can be found by subtracting the area of the picture from the total area:
\[
\text{Area of the border} = \text{Area of the entire framed picture} - \text{Area of the picture} = 96 \, \text{square inches} - 60 \, \text{square inches} = 36 \, \text{square inches}
\]
4. Let \( x \) be the width of the border. The dimensions of the framed picture (including the border) will be:
- Length: \( 10 + 2x \)
- Width: \( 6 + 2x \)
5. The area of the framed picture can be expressed as:
\[
(10 + 2x)(6 + 2x) = 96
\]
6. Expanding this equation:
\[
60 + 20x + 12x + 4x^2 = 96
\]
\[
4x^2 + 32x + 60 = 96
\]
\[
4x^2 + 32x - 36 = 0
\]
7. We can simplify the equation by dividing everything by 4:
\[
x^2 + 8x - 9 = 0
\]
8. Now, we can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} = \frac{-8 \pm \sqrt{64 + 36}}{2}
\]
\[
= \frac{-8 \pm \sqrt{100}}{2} = \frac{-8 \pm 10}{2}
\]
9. This gives us two possible solutions:
\[
x = \frac{2}{2} = 1 \quad \text{(positive solution)}
\]
\[
x = \frac{-18}{2} \quad \text{(negative solution, not valid)}
\]
Therefore, the width of the border is **1 inch**.
The correct answer is **1 inch**.
1. The area of the picture itself is calculated as follows:
\[
\text{Area of the picture} = \text{length} \times \text{width} = 10 \, \text{inches} \times 6 \, \text{inches} = 60 \, \text{square inches}
\]
2. The area of the entire framed picture is given as 96 square inches.
3. The area of the border can be found by subtracting the area of the picture from the total area:
\[
\text{Area of the border} = \text{Area of the entire framed picture} - \text{Area of the picture} = 96 \, \text{square inches} - 60 \, \text{square inches} = 36 \, \text{square inches}
\]
4. Let \( x \) be the width of the border. The dimensions of the framed picture (including the border) will be:
- Length: \( 10 + 2x \)
- Width: \( 6 + 2x \)
5. The area of the framed picture can be expressed as:
\[
(10 + 2x)(6 + 2x) = 96
\]
6. Expanding this equation:
\[
60 + 20x + 12x + 4x^2 = 96
\]
\[
4x^2 + 32x + 60 = 96
\]
\[
4x^2 + 32x - 36 = 0
\]
7. We can simplify the equation by dividing everything by 4:
\[
x^2 + 8x - 9 = 0
\]
8. Now, we can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} = \frac{-8 \pm \sqrt{64 + 36}}{2}
\]
\[
= \frac{-8 \pm \sqrt{100}}{2} = \frac{-8 \pm 10}{2}
\]
9. This gives us two possible solutions:
\[
x = \frac{2}{2} = 1 \quad \text{(positive solution)}
\]
\[
x = \frac{-18}{2} \quad \text{(negative solution, not valid)}
\]
Therefore, the width of the border is **1 inch**.
The correct answer is **1 inch**.