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Todd has a rectangular garden with an area of 30 square feet. If the garden is 13 feet longer...

Question:

Todd has a rectangular garden with an area of 30 square feet.

If the garden is 13 feet longer than it is wide, what is the perimeter of the garden (in feet)?

Rectangular Area:

Area is the amount of space that is covered by a given geometrical figure or object. A rectangle is a two-dimensional figure that has a length and a width. The rectangular area is the area that is covered by the rectangular figure. We calculate the area of a rectangle using the formula {eq}A = l\times w {/eq}.

Answer and Explanation:


Let the width of the rectangular garden be {eq}w\, ft {/eq}. If the length is 13ft longer than the width, then the expression for the length is:

  • {eq}l = (w + 13)\, ft {/eq}

The rectangular garden has an area of {eq}A = 30\, ft^2 {/eq}. Therefore:

  • {eq}30 = w(w + 13) {/eq}
  • {eq}30 = w^2 + 13w {/eq}
  • {eq}w^2 + 13w - 30 = 0 {/eq}

Solving the quadratic equation by the use of factorization, we have:

  • {eq}w^2 + 15w - 2w - 30 = 0 {/eq}
  • {eq}w(w + 15) - 2(w + 15) = 0 {/eq}
  • {eq}(w - 2)(w + 15) = 0 {/eq}
  • {eq}w = 2\, ft,\quad w = -15\, ft {/eq}

Since length cannot be negative, we will consider the positive value of w.

Therefore, the garden measurements are:

  • {eq}w = 2\, ft \,\rm and\, l = 2 + 13 = 15\, ft {/eq}.

The perimeter of a rectangle is given by:

  • {eq}P = 2(w + l) {/eq}

Thus, the perimeter of the rectangular garden is:

  • {eq}P = 2(2 + 15) = \boxed{34\, ft} {/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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