Final answer:
The dimensions of the larger square garden are found by setting up equations using the perimeter and the area in relation to the smaller garden. Solving the equations reveals the larger garden has a side length of 19 meters, a perimeter of 76 meters, and an area of 361 square meters.
Step-by-step explanation:
The problem concerns finding the dimensions of a larger square garden given its perimeter and area in comparison to a smaller square garden. Let's denote the side of the smaller square as s meters. Since it is a square, all sides are equal, thus its perimeter is 4s meters. For the larger square, its perimeter is 12 meters greater than that of the smaller one, so its perimeter is 4s + 12 meters and its side length will be s + 3 meters, dividing the perimeter by 4.
The area of a square is given by side length squared, so the area of the smaller square is s^2 and the area of the larger one is (s+3)^2. We're told that the area of the larger garden is 105 square meters greater than the smaller garden, giving us the equation (s+3)^2 = s^2 + 105. Expanding the left side of the equation yields s^2 + 6s + 9, and subtracting s^2 from both sides we get 6s + 9 = 105. Solving for s gives us s = 16 meters.
The dimensions of the larger garden therefore can be calculated as the side of the smaller garden (16 meters) plus 3 meters, resulting in a side length of 19 meters for the larger garden. The perimeter of the larger garden will be 4 x 19 = 76 meters, and the area will be 19^2 = 361 square meters.