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The length of a rectangular vegetable garden is 4 feet more than its width. After a 2-foot cement border is placed around the garden, the area of garden and border is 320 square feet.

a. Make an illustration of the vegetable garden.

b. Find the original dimensions of the vegetable garden.



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KAntoineDoixidn

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[tex] \large\underline{\mathbb{PROBLEM}:} [/tex]

» The length of a rectangular vegetable garden is 4 feet more than its width. After a 2-foot cement border is placed around the garden, the area of garden and border is 320 square feet.

  • a. Make an illustration of the vegetable garden.

  • b. Find the original dimensions of the vegetable garden.

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[tex] \large\underline{\mathbb{ANSWER}:} [/tex]

[tex] \qquad \Large \:\: \rm{length = 16 \: feet} [/tex]

[tex] \qquad \Large \:\: \rm{width = 12 \: feet} [/tex]

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[tex] \large\underline{\mathbb{SOLUTION}:} [/tex]

» Let l and w be the length and the width of the garden. Make the length as 4 more than the width.

  • [tex] l = w + 4 [/tex]

» With a 2 feet thick border. The length and the width of the garden with the border would be.

  • [tex] length = l + 2 + 2 = l + 4 [/tex]

  • [tex] width = w + 2 + 2 = w + 4 [/tex]

» Create two equations by the given statements.

  • [tex] \begin{cases} l = w + 4 \\ (l + 4)(w + 4) = 320 \end{cases} \quad \begin{align} \tt{(eq. \: 1)} \\ \tt{(eq. \: 2)} \end{align} [/tex]

» Substitute l in the second equation from the first equation in terms of w.

  • [tex] \begin{cases} l = w + 4 \\ (w + 4 + 4)(w + 4) = 320 \end{cases}[/tex]

  • [tex] \begin{cases} l = w + 4 \\ (w + 8)(w + 4) = 320 \end{cases}[/tex]

  • [tex] \begin{cases} l = w + 4 \\ w^2 + 12w + 32 = 320 \end{cases}[/tex]

  • [tex] \begin{cases} l = w + 4 \\ w^2 + 12w + 32 - 320 = 0 \end{cases}[/tex]

  • [tex] \begin{cases} l = w + 4 \\ w^2 + 12w - 288 = 0 \end{cases}[/tex]

» Solve the quadratic equation in the second equation by factoring. Use only the positive solution.

  • [tex] w^2 + 12w - 288 = 0 [/tex]

  • [tex] (w + 24)(w - 12) = 0 [/tex]

  • [tex] w + 24 = 0 \quad,\quad w - 12 = 0 [/tex]

  • [tex] w = \text-24 \quad,\quad \boxed{w = 12} [/tex]

» Thus, the width of the garden is 12 feet. Substitute it to the first equation to find the length.

  • [tex] \begin{cases} l = 12 + 4 \\ w = 12 \end{cases}[/tex]

  • [tex] \begin{cases} l = 16 \\ w = 12 \end{cases}[/tex]

[tex] \therefore [/tex] The length of the garden is 16 feet in measure.

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