A rectangular picture frame contains a 12 inch x 14 inch picture. What is the width of the frame...

Question:

A rectangular picture frame contains a 12 inch x 14 inch picture.

What is the width of the frame if the total area of the picture and frame is 288 square inches?

The Area of a Rectangle:

A rectangle is a closed plane figure that has four sides. The opposite sides of a rectangle are congruent, but the adjacent sides are not. The area of a rectangle is the amount of space that is covered by the rectangular figure, and it is computed using the formula {eq}A = l\times w {/eq}.

Answer and Explanation:


Let the width of the frame be {eq}x\; \rm in {/eq}. If the length of the picture is {eq}l = 14\; \rm in {/eq} and the width is {eq}w = 12 \; \rm in {/eq}, then the dimensions of the picture frame plus the picture is:

  • {eq}l_1 = 14 + 2x {/eq}
  • {eq}w_1 = 12 + 2x {/eq}

The area of the picture frame plus the picture will, therefore, be expressed as:

  • {eq}A = l_1\times w_1 {/eq}
  • {eq}A = (14 + 2x)(12 + 2x) {/eq}

If the total area of the picture frame plus the picture is 288 square inches, then:

  • {eq}288 = (14 + 2x)(12 + 2x) {/eq}

Expanding the RHS of the equation:

  • {eq}288 = 168 + 52x + 4x^2 {/eq}
  • {eq}120 = 4x^2 + 52x {/eq}
  • {eq}30 = x^2 + 13x {/eq}
  • {eq}x^2 + 13x - 30 = 0 {/eq}

We have a quadratic equation to solve. Factoring the quadratic equation, we get:

  • {eq}x^2 + 15x - 2x - 30 = 0 {/eq}
  • {eq}x(x + 15) - 2(x + 15) = 0 {/eq}
  • {eq}(x - 2)(x + 15) = 0 {/eq}

Therefore:

  • {eq}\rm x = -15\; in, \quad x = 2\; in {/eq}

Considering the positive value of x, we have:

  • {eq}\boxed{x = 2\; \rm in} {/eq}

Thus, the width of the picture frame is 2 inches.


Learn more about this topic:

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Measuring the Area of a Rectangle: Formula & Examples

from Geometry: High School

Chapter 8 / Lesson 7
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