Holly has a rectangular garden that measures 12m by 14m. SHe wants to increase the area to 255m^2 by increasing the width and the length by the same amount. What will the dimensions be of the new garden ?
-solving that involves quadratic equations
New width = 12 + x
New length = 14 + x
Area = (12 + x)(14 + x) = 225
To solve this problem, let's assume that Holly wants to increase the dimensions of the garden by adding "x" to both the width and length.
The original dimensions of the garden are 12m by 14m, and the original area is given by:
Original area = length * width = 12m * 14m = 168m^2
Holly wants to increase the area to 255m^2, so the new area is given by:
New area = (length + x) * (width + x) = 255m^2
We can set up a quadratic equation to solve for "x". Expanding the equation above, we get:
(length + x) * (width + x) = 255m^2
(length * width) + (length * x) + (width * x) + (x * x) = 255m^2
Since the original area is 168m^2, we can substitute the values and simplify the equation:
168 + 12x + 14x + x^2 = 255
x^2 + 26x + 168 - 255 = 0
x^2 + 26x - 87 = 0
We can now solve this quadratic equation to find the value of "x". Factoring or using the quadratic formula will give us the solution. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this equation, a = 1, b = 26, and c = -87. Plugging the values into the quadratic formula, we get:
x = (-26 ± √(26^2 - 4(1)(-87))) / 2(1)
x = (-26 ± √(676 + 348)) / 2
x = (-26 ± √1024) / 2
x = (-26 ± 32) / 2
We get two possible solutions for "x":
x1 = (-26 + 32) / 2 = 3
x2 = (-26 - 32) / 2 = -29
Since we can't have a negative dimension for the garden, we consider only the positive solution, x = 3.
Therefore, the new dimensions of the garden with increased area will be:
Width + x = 12m + 3m = 15m
Length + x = 14m + 3m = 17m
So, the new garden will have dimensions of 15m by 17m.
To solve this problem, we need to find the dimensions of the new garden when the area is increased to 255m².
Let's assume that Holly increases both the length and the width by the same amount, let's say 'x' meters.
The original area of the rectangular garden is given by the length multiplied by the width:
Original Area = Length * Width
= 12m * 14m
= 168m²
The new area of the garden after increasing the dimensions by 'x' meters will be:
New Area = (Length + x) * (Width + x)
= (12m + x) * (14m + x)
= 168m² + 12m * x + 14m * x + x²
= 168m² + 26m * x + x²
Since we know that the new area is 255m², we can set up a quadratic equation and solve for 'x'.
255m² = 168m² + 26m * x + x²
Simplifying the equation by subtracting 168m² from both sides:
x² + 26m * x - 87m² = 0
Now we have a quadratic equation in terms of 'x'. To solve this equation, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In our case, a = 1, b = 26m, and c = -87m².
Substituting the values into the quadratic formula:
x = (-(26m) ± √((26m)² - 4(1)(-87m²))) / (2(1))
Simplifying further:
x = (-26m ± √(676m² + 348m²)) / 2
x = (-26m ± √1024m²)/2
x = (-26m ± 32m)/2
Now we have two possibilities for 'x' when solving the quadratic equation, which are:
x₁ = (-26m + 32m)/2 = 6m/2 = 3m
x₂ = (-26m - 32m)/2 = -58m/2 = -29m
Since we are looking for positive dimensions for the garden, we can ignore the negative value of 'x'. Therefore, the increase in dimensions would be 3m.
Finally, to find the new dimensions, we add 'x' to both the length and width of the original garden:
New Length = 12m + 3m = 15m
New Width = 14m + 3m = 17m
So, the dimensions of the new garden will be 15m by 17m.