To find the dimensions of the rectangular vegetable garden, we need to solve the equation:
length * width = 180.
We also know that the length is 3 feet longer than the width, so
length = width + 3.
Substituting the second equation into the first equation, we get:
(width + 3) * width = 180.
Expanding the equation, we get:
width^2 + 3width = 180.
Rearranging the equation to a quadratic form, we get:
width^2 + 3width - 180 = 0.
We can factor this equation as:
(width + 15)(width - 12) = 0.
This gives us two possible solutions:
width + 15 = 0 or width - 12 = 0.
Solving for width, we get:
width = -15 or width = 12.
Since a negative width doesn't make sense in this context, we can ignore the first solution.
Therefore, the width of the vegetable garden is 12 feet.
Substituting this value back into the second equation, we find the length:
length = width + 3 = 12 + 3 = 15 feet.
Therefore, the correct dimensions for the vegetable garden are 12 feet by 15 feet.