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 dimensions=15&17
(I4 x)(12 x)=255
168 14x 12x x^2=255
168 26x x^2=255
x^2 26x-57=0..then solve as a quadratic equation
To solve this problem, we can use quadratic equations. Let's start by finding the area of the current garden:
Area = Length * Width = 12m * 14m = 168m²
We need to increase the area to 255m². Let's assume that both the length and the width will be increased by the same amount, which we'll call 'x'. Therefore, the new length will be 12m + x, and the new width will be 14m + x.
The area of the new garden can be calculated as:
New Area = (12m + x) * (14m + x)
Setting up the equation:
(12m + x) * (14m + x) = 255m²
Expanding this equation:
168m² + 12m * x + 14m * x + x² = 255m²
Rearranging the equation and simplifying:
x² + 26x - 87 = 0
Now, we have a quadratic equation: ax² + bx + c = 0, where:
a = 1
b = 26
c = -87
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Plugging in the values:
x = (-26 ± √(26² - 4 * 1 * -87)) / 2 * 1
Simplifying:
x = (-26 ± √(676 + 348)) / 2
x = (-26 ± √1024) / 2
x = (-26 ± 32) / 2
Now we have two possible solutions for x:
x₁ = (-26 + 32) / 2 = 3
x₂ = (-26 - 32) / 2 = -29
Since the dimensions cannot be negative, we discard the negative solution. Therefore, the value of x is 3.
To find the new dimensions of the garden, we can substitute x = 3 back into the expressions for length and width:
New length = 12m + x = 12m + 3m = 15m
New width = 14m + x = 14m + 3m = 17m
So, the new dimensions of the garden will be 15m by 17m.