Math Problem Statement

A 5 inch × 7 inch photograph is placed inside a picture frame. Both the length and width of the frame are 2x inches larger than the width and length of the photograph. Which expression represents the perimeter of the frame?

Solution

Let's solve this step by step:

Given:

  • The dimensions of the photograph are 55 inches by 77 inches.
  • The frame adds 2x2x inches to both the length and width of the photograph.

Find:

The expression for the perimeter of the frame.

Step 1: Determine the dimensions of the frame

  • Length of the frame: The length of the photograph is 77 inches. Since the frame adds 2x2x inches to this dimension, the length of the frame is: 7+2x inches7 + 2x \text{ inches}

  • Width of the frame: The width of the photograph is 55 inches. Since the frame adds 2x2x inches to this dimension, the width of the frame is: 5+2x inches5 + 2x \text{ inches}

Step 2: Expression for the perimeter

The perimeter PP of a rectangle is given by: P=2×(Length + Width)P = 2 \times \text{(Length + Width)}

Substitute the values for the length and width: P=2×((7+2x)+(5+2x))P = 2 \times \left( (7 + 2x) + (5 + 2x) \right)

Step 3: Simplify the expression

Combine the like terms inside the parentheses: P=2×(7+5+2x+2x)P = 2 \times \left( 7 + 5 + 2x + 2x \right) P=2×(12+4x)P = 2 \times \left( 12 + 4x \right)

Finally, distribute the 2: P=24+8xP = 24 + 8x

Answer:

The expression that represents the perimeter of the frame is 24+8x24 + 8x.

Would you like more details or have any further questions?

Here are five related questions to explore:

  1. How do you find the area of the frame, given these dimensions?
  2. What happens to the perimeter if xx doubles?
  3. How does the expression for the perimeter change if the frame adds 3x3x inches instead of 2x2x inches?
  4. How would you represent the area of the frame in terms of xx?
  5. How do the length and width affect the perimeter of a rectangle?

Tip: When dealing with problems involving rectangles, it's useful to remember that both the length and width contribute equally to the perimeter calculation.

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Math Problem Analysis

Mathematical Concepts

Geometry
Perimeter of Rectangle

Formulas

Perimeter of Rectangle

Theorems

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Suitable Grade Level

Grades 6-8