New Clarendon Park is undergoing renovations to its gardens. One garden that was originally a square is being adjusted so that o
ne side is doubled in length, while the other side is decreased by three meters. The new rectangular garden will have an area that is 25% more that the original square garden. Write an equation that could be used to determine the length of a side of the original square garden.
Determine the area, in square meters, of the new rectangular garden.
The square area is expressed as: A = a², where A is the area of the square, and a is the side of the square.
The rectangle area is expressed as: A₁ = a₁ · b₁, where A₁ is the area of the rectangle, and a₁ and b₁ are the sides of the rectangle.
After renovations, square garden becomes rectangular.
One side is doubled in length: a₁ = 2a
The other side is decreased by three meters. b₁ = a - 3
The new area is 25% than the original square garden: A₁ = A + 25%A = = A + 25/100·A = A + 1/25·A = a² + 1/25·a² = <span>a² + 0.25·a² </span> = 1.25·a²
If the starting equation is: A₁ = a₁ · b₁
Thus, the equation is: 1.25a² = 2a·(<span>a - 3) </span>1.25a² = 2a · a - 2a · 3 1.25a² = 2a² - 6a
<span>Therefore, the equation that could be used to determine the length of a side of the original square garden is: </span><u>2a² - 6a = </u><span><u>1.25a²</u></span>
Now, we will solve the equation: 2a² - 6a = 1.25a² 2a² - 1.25a² - 6a = 0 0.75a² - 6a = 0 ⇒ a(0.75a - 6) = 0
From here, one of the multiplier must be zero - either a or (0.75a - 6). Since a could not be zero, (0.75a - 6) is: 0.75a - 6 = 0 0.75a = 6 a = 6 ÷ 0.75 a = 8
If the side of the square is 8, then the area of the rectangle is A₁ = 1.25 · a² A₁ = 1.25 ·8² A₁ = 1.25 · 64 A₁ = 80
Therefore, the area of the new rectangle garden is 80 square meters.
