A rectangular garden has a walk around it of a width x. The garden is 20 ft by 15 ft. Write a...

Question:

A rectangular garden has a walk around it of a width x. The garden is 20 ft by 15 ft. Write a function representing the combined area, A(x), of the garden walk. Write as a polynomial in standard form. Use the function you came up with in the previous question to find the combined area if the walkway is 3ft. Write a sentence that gives your labeled answer.

The Area of a Rectangle:

A rectangle is a closed plane figure that has four line segments for sides. The opposite sides of a rectangle are congruent, and the interior angles are right angles. The area of a rectangle is the amount of space that a rectangular figure covers. If a rectangle measures {eq}l {/eq} units by {eq}w {/eq} units, then its area is calculated as {eq}A = l\times w {/eq} square units.

Answer and Explanation:


(a). Write a function representing the combined area, A(x), of the garden walk. Write as a polynomial in standard form.


Let's start by calculating the area of the garden. The garden measures 20ft by 15ft. Its area is equal to:

  • {eq}A = \rm 20ft \times 15ft = 300\, \rm ft^2 {/eq}

The width of the walk is {eq}\rm x\, ft {/eq}. The combined area will, therefore, be equal to:

  • {eq}A(x) = (20 + 2x)(15 + 2x) {/eq}
  • {eq}A(x) = 300 + 70x + 4x^2 {/eq}
  • {eq}A(x) = \boxed{4x^2 + 70x + 300} {/eq}

The equation above shows the combined area of the garden and the walk.


(b). Use the function you came up with in the previous question to find the combined area if the walkway is 3ft. Write a sentence that gives your labeled answer.

If the walk measures {eq}x = 3\rm ft {/eq}, the combined area is equal to:

  • {eq}A(x) = 4(3)^2 + 70(3) + 300 = \boxed{546\, \rm ft^2} {/eq}

The combined area of the garden measuring 20ft by 15ft and a walk of width 3ft around it is equal to 546 square feet.


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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