A gardener wants to create a rectangular vegetable garden in a backyard. She wants to have a total area of 13 square feet, and it should be 12 feet longer than it is wide. What dimensions should she use for the vegetable garden?
width = feet
length = feet
To find the dimensions of the rectangular vegetable garden, we need to solve the given information.
Let's assume the width of the vegetable garden is x feet.
According to the given information, the length of the vegetable garden is 12 feet longer than the width. So, the length would be (x + 12) feet.
To find the dimensions that satisfy the condition, we need to find the values of x and (x + 12) that satisfy the equation for the total area.
The total area of a rectangle is given by multiplying its length and width.
So, the equation for the total area can be expressed as:
x * (x + 12) = 13
Now, we can solve this equation to find the width (x) and length (x + 12) of the vegetable garden.
Expanding the equation:
x^2 + 12x = 13
Rearranging the equation:
x^2 + 12x - 13 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula.
To simplify the calculation, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation x^2 + 12x - 13 = 0, a = 1, b = 12, and c = -13.
Plugging in these values into the quadratic formula:
x = (-12 ± √(12^2 - 4(1)(-13))) / (2*1)
Simplifying the equation further:
x = (-12 ± √(144 + 52)) / 2
x = (-12 ± √196) / 2
x = (-12 ± 14) / 2
Solving for both values of x:
x = (-12 + 14) / 2 = 1
x = (-12 - 14) / 2 = -13
Since the width cannot be negative, we can discard the negative value. Hence, the width of the vegetable garden is 1 foot.
Now, to find the length, we can substitute this value back into the expression (x + 12):
Length = 1 + 12 = 13 feet
Therefore, the dimensions of the vegetable garden should be:
Width = 1 foot
Length = 13 feet