Question:
The figure above represents a square photograph bordered by a frame that has a uniform width of 3 inches. If the frame and the picture have the same area, and each of the photograph's sides measures x inches, which of the following equations is true?
The figure above represents a square photograph bordered by a frame that has a uniform width of 3 inches. If the frame and the picture have the same area, and each of the photograph's sides measures x inches, which of the following equations is true?
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When dealing with frames or borders, remember that the width is added to both sides of the inner object. A common mistake is to add the width only once (e.g., using \(x+3\) instead of \(x+6\)). Visualizing or drawing a simple diagram can help prevent this error.
Updated On: Oct 3, 2025
- \(x^2 = 6x + 9\)
- \(x^2 = 3x^2 + 2x\)
- \(x^2 = 12x + 36\)
- \(x^2 = 6x + 36\)
- \(x^2 = 6x^2 + 4x\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
The problem asks us to set up an equation based on the geometric properties and areas of a square photograph and its surrounding frame. We need to express the areas of the photograph and the frame in terms of the variable \(x\) and then set them equal to each other.
Step 2: Key Formula or Approach:
1. Area of the square photograph: \(A_{\text{photo}} = \text{side}^2 = x^2\). 2. Determine the dimensions of the entire object (photograph + frame). The frame adds 3 inches on all four sides. So, the total side length is \(x + 3 + 3 = x + 6\). 3. Area of the entire object: \(A_{\text{total}} = (x+6)^2\). 4. Area of the frame: \(A_{\text{frame}} = A_{\text{total}} - A_{\text{photo}}\). 5. Set \(A_{\text{frame}} = A_{\text{photo}}\) and simplify.
Step 3: Detailed Explanation:
The area of the square photograph with side length \(x\) is: \[ A_{\text{photo}} = x^2 \] The total side length of the photograph with the frame is \(x+6\) inches. The total area is: \[ A_{\text{total}} = (x+6)^2 = x^2 + 12x + 36 \] The area of the frame is the total area minus the area of the photograph: \[ A_{\text{frame}} = A_{\text{total}} - A_{\text{photo}} = (x^2 + 12x + 36) - x^2 = 12x + 36 \] The problem states that the frame and the picture have the same area: \[ A_{\text{frame}} = A_{\text{photo}} \] Substituting the expressions we found: \[ 12x + 36 = x^2 \] Step 4: Final Answer:
The equation relating the areas is \(x^2 = 12x + 36\). This matches option (C).
The problem asks us to set up an equation based on the geometric properties and areas of a square photograph and its surrounding frame. We need to express the areas of the photograph and the frame in terms of the variable \(x\) and then set them equal to each other.
Step 2: Key Formula or Approach:
1. Area of the square photograph: \(A_{\text{photo}} = \text{side}^2 = x^2\). 2. Determine the dimensions of the entire object (photograph + frame). The frame adds 3 inches on all four sides. So, the total side length is \(x + 3 + 3 = x + 6\). 3. Area of the entire object: \(A_{\text{total}} = (x+6)^2\). 4. Area of the frame: \(A_{\text{frame}} = A_{\text{total}} - A_{\text{photo}}\). 5. Set \(A_{\text{frame}} = A_{\text{photo}}\) and simplify.
Step 3: Detailed Explanation:
The area of the square photograph with side length \(x\) is: \[ A_{\text{photo}} = x^2 \] The total side length of the photograph with the frame is \(x+6\) inches. The total area is: \[ A_{\text{total}} = (x+6)^2 = x^2 + 12x + 36 \] The area of the frame is the total area minus the area of the photograph: \[ A_{\text{frame}} = A_{\text{total}} - A_{\text{photo}} = (x^2 + 12x + 36) - x^2 = 12x + 36 \] The problem states that the frame and the picture have the same area: \[ A_{\text{frame}} = A_{\text{photo}} \] Substituting the expressions we found: \[ 12x + 36 = x^2 \] Step 4: Final Answer:
The equation relating the areas is \(x^2 = 12x + 36\). This matches option (C).
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