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A large square portrait is framed. The frame(border) is a unifrom length of 1.4m wide. If the...

Question:

A large square portrait is framed. The frame(border) is a unifrom length of 1.4m wide. If the area of the picture equals the area of the frame, find the dimesions of the picture.

The Area of a Square:

A square is a figure in two dimensional that has two pairs of parallel sides. All the sides in of a square are equal, and each of the interior angles is a right angle. The area of a square is the amount of space covered by the figure. We calculate the square area using the formula {eq}A = l^2 {/eq}, where {eq}l {/eq} represents the side length of the square.

Answer and Explanation:


The area of a square is given by:

  • {eq}A = l^2 {/eq}

Where {eq}l {/eq} is the length.

Let the length of the frame be {eq}x\; \rm m {/eq}. If the width of the border is {eq}1.4\; \rm m {/eq}, then the length of the square picture is:

  • {eq}a = l - 1.4 - 1.4 {/eq}
  • {eq}a = l - 2.8 {/eq}

The area of the border will be given by:

  • {eq}A_1 = l^2 - (l - 2.8)^2 {/eq}

And the area of the picture is equal to:

  • {eq}A_2 = (l - 2.8)^2 {/eq}

If the area of the picture is equal to the area of the border, then:

  • {eq}A_1 = A_2 {/eq}
  • {eq}l^2 - (l - 2.8)^2 = (l - 2.8)^2 {/eq}
  • {eq}l^2 = 2 (l - 2.8)^2 {/eq}

Expanding the RHS of the equation:

  • {eq}l^2 = 2l^2 - 11.2l + 15.68 {/eq}
  • {eq}l^2 - 11.2l + 15.68= 0 {/eq}

Solving the quadratic equation:

  • {eq}l = \dfrac{-b\pm \sqrt{b^2 - 4ac}}{2a} {/eq}

{eq}a = 1,\; b = -11.2\; c = 15.68 {/eq}

Therefore:

  • {eq}l = \dfrac{11.2\pm \sqrt{125.44 - 4(1)(15.68)}}{2} {/eq}
  • {eq}l = \dfrac{11.2\pm \sqrt{62.72}}{2} {/eq}
  • {eq}l = \dfrac{11.2\pm 7.9196}{2} {/eq}
  • {eq}l = 5.6 \pm 3.9598 {/eq}
  • {eq}l \approx 9.56\; \rm m, \quad l \approx 1.64\; \rm m {/eq}

Therefore:

  • {eq}a= 9.56 - 2.8 = 6.76\; \rm m {/eq}
  • {eq}a= 1.64 - 2.8 = -1.16\; \rm m {/eq}

Since the length of the picture cannot be negative, then the dimensions of the picture are:

  • {eq}\boxed{\color{blue}{6.76\; \rm m \; by\; 6.76\; \rm m }} {/eq}

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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