# 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.

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from Geometry: High School

Chapter 8 / Lesson 7