$8$ . The community garden club has a vegetable garden that measures $15 \mathrm{~m}$ by $30 \mathrm{~m}$ . One of the members has donated a new piece of land for a larger garden. They plan to increase the garden by $250 \mathrm{~m}^{2}$ . However, because of the dimensions of the new land, both dimensions of the original garden must be increased by the same amount. Determine the dimensions of the new garden. $\newline$ $\text { [20 m } 35 \mathrm{~m} \text { ] }$

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8

. The community garden club has a vegetable garden that measures

15 \mathrm{~m}

30 \mathrm{~m}

. One of the members has donated a new piece of land for a larger garden. They plan to increase the garden by

250 \mathrm{~m}^{2}

. However, because of the dimensions of the new land, both dimensions of the original garden must be increased by the same amount. Determine the dimensions of the new garden.

\newline

\text { [20 m } 35 \mathrm{~m} \text { ] }

Calculate Original Area: The original area of the garden is $15\,\text{m}$ by $30\,\text{m}$ , so the original area is $15\,\text{m} \times 30\,\text{m}$ . $\newline$ Calculate the original area: $15\,\text{m} \times 30\,\text{m} = 450\,\text{m}^2$ .
Calculate New Area: The new area will be the original area plus the increase of $250\,\text{m}^2$ . $\newline$ Calculate the new area: $450\,\text{m}^2 + 250\,\text{m}^2 = 700\,\text{m}^2$ .
Write Equation for New Area: Let $x$ be the amount by which each dimension is increased. The new dimensions will be $(15m + x)$ and $(30m + x)$ . Write the equation for the new area: $(15m + x)(30m + x) = 700m^2$ .
Expand and Simplify Equation: Expand the equation: $450m^2 + 15mx + 30mx + x^2 = 700m^2$ . Simplify the equation: $x^2 + 45mx - 250m^2 = 0$ .
Use Quadratic Formula: Use the quadratic formula to solve for $x$ : $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 1$ , $b = 45m$ , and $c = -250m^2$ . Calculate the discriminant: $(45m)^2 - 4(1)(-250m^2) = 2025m^2 + 1000m^2 = 3025m^2$ .
Calculate Discriminant: Calculate the square root of the discriminant: $\sqrt{3025m^2} = 55m$ . $\newline$ Plug into the quadratic formula: $x = \frac{-45m \pm 55m}{2}$ .
Calculate Square Root: There are two possible solutions for $x$ : $x = 5m$ or $x = -50m$ . Since a negative increase in dimensions doesn't make sense, we discard $x = -50m$ .
Find Possible Solutions: The increase in each dimension is $5\,\text{m}$ . So the new dimensions are $15\,\text{m} + 5\,\text{m}$ and $30\,\text{m} + 5\,\text{m}$ . Calculate the new dimensions: $20\,\text{m}$ and $35\,\text{m}$ .