Certainly! Let's solve each part of the question step-by-step.
### (i) Total Number of Rose Plants in the Garden:
Given that there are 41 rose plants in each row and each column of a square garden, the total number of rose plants can be calculated as follows:
Since the garden is square and each side has 41 plants, the total number of plants is:
[tex]\[ \text{Total plants} = 41 \times 41 = 1681 \][/tex]
So, there are 1681 rose plants in the garden.
### (ii) Area of the Garden:
To find the area of the garden, we need to determine the side length of the square garden.
Given the distance between each pair of rose plants is 1 ft, consider the spacing:
1. For 41 plants, there are 40 gaps (since the spacing is between the plants).
2. Therefore, the side length of the garden is 40 ft.
The area of a square garden can be found using the formula:
[tex]\[ \text{Area} = \text{side length}^2 \][/tex]
[tex]\[ \text{Area of the garden} = 40 \times 40 = 1600 \, \text{square feet} \][/tex]
So, the area of the garden when there are 41 rose plants in each row and column is 1600 square feet.
### (iii) Additional Area Required for New Garden:
Now, we need to determine how much more area is required if we modify the garden to plant 40 rose plants in each row and column, assuming the distance between each pair of rose plants remains 1 ft.
Again, considering the gaps:
1. For 40 plants, there are 39 gaps.
2. Therefore, the new side length of the garden is 39 ft.
The area of the new garden can be calculated as:
[tex]\[ \text{New Area} = 39 \times 39 = 1521 \, \text{square feet} \][/tex]
To find the additional area required, we subtract the area of the original garden from the area of the new garden:
[tex]\[ \text{Additional Area} = \text{New Area} - \text{Original Area} \][/tex]
[tex]\[ \text{Additional Area} = 1521 - 1600 = -79 \][/tex]
Since this value is negative, it actually indicates a reduction in area rather than additional area required. If the goal was to get more area, we would need to reconsider the requirements or conditions.
Hence, changing the number of rose plants from 41 to 40 per row and column would actually reduce the area by 79 square feet.
