Question

На участке прямоугольной формы со сторонами 7 м и 6 м хотят разместить прямоугольную клумбу площадью 12 м в квадрате так , чтобы вокруг клумбы образовалась дорожка одинаковой ширины. Какую ширину должна иметь дорожка ? ( С решением полностью )

Answer 1

Если методом подбора, то пробуем сделать дорожку в 1 м  = 6 и 5 метров (площадь - 30 кв.м), дорожку в 2 м  = 5 и 4 метров (площадь - 20 кв. м)
дорожку в 3 м , тогда посередине  клумба будет иметь размеры 7-3 = 4 м и 6-3 = 3 м Площадь клумбы притаких размерах - 4 х 3 = 12 кв. м.
Т.е. дорожка будет шириной 3 метра

Answer 2

Problem Analysis

We have a rectangular plot of land with dimensions 7m by 6m. We want to place a rectangular flower bed with an area of 12 square meters in such a way that there is an equal-width pathway around the flower bed. We need to determine the width of the pathway.

Solution

To find the width of the pathway, we can subtract the area of the flower bed from the total area of the plot and then divide by the perimeter of the flower bed to get the width of the pathway.

Let's calculate the width of the pathway step by step:

1. Calculate the area of the plot: - The area of the plot is equal to the product of its length and width. - The plot has dimensions 7m by 6m, so its area is 7m * 6m = 42 square meters.

2. Calculate the area of the flower bed: - The area of the flower bed is given as 12 square meters.

3. Calculate the area of the pathway: - The area of the pathway is equal to the difference between the area of the plot and the area of the flower bed. - Area of the pathway = Area of the plot - Area of the flower bed = 42 square meters - 12 square meters = 30 square meters.

4. Calculate the perimeter of the flower bed: - The perimeter of a rectangle is equal to twice the sum of its length and width. - The flower bed has an area of 12 square meters, so we need to find the dimensions of the rectangle with an area of 12 square meters. - Let's assume the length of the flower bed is L meters and the width is W meters. - We have the equation L * W = 12. - We can solve this equation to find the dimensions of the flower bed. - Since the flower bed is rectangular, we can assume L > W. - Let's try different values of L and W that satisfy the equation L * W = 12: - L = 1m, W = 12m (not possible because L > W) - L = 2m, W = 6m (possible) - L = 3m, W = 4m (possible) - L = 4m, W = 3m (possible) - L = 6m, W = 2m (not possible because L > W) - L = 12m, W = 1m (not possible because L > W) - So, the possible dimensions of the flower bed are 2m by 6m, 3m by 4m, and 4m by 3m.

5. Calculate the perimeter of the flower bed for each possible dimension: - For a flower bed with dimensions 2m by 6m: - Perimeter = 2 * (Length + Width) = 2 * (2m + 6m) = 2 * 8m = 16m. - For a flower bed with dimensions 3m by 4m: - Perimeter = 2 * (Length + Width) = 2 * (3m + 4m) = 2 * 7m = 14m. - For a flower bed with dimensions 4m by 3m: - Perimeter = 2 * (Length + Width) = 2 * (4m + 3m) = 2 * 7m = 14m.

6. Calculate the width of the pathway: - The width of the pathway is equal to the area of the pathway divided by the perimeter of the flower bed. - For a flower bed with dimensions 2m by 6m: - Width of the pathway = Area of the pathway / Perimeter of the flower bed = 30 square meters / 16m = 1.875m. - For a flower bed with dimensions 3m by 4m: - Width of the pathway = Area of the pathway / Perimeter of the flower bed = 30 square meters / 14m = 2.143m. - For a flower bed with dimensions 4m by 3m: - Width of the pathway = Area of the pathway / Perimeter of the flower bed = 30 square meters / 14m = 2.143m.

Answer

The width of the pathway should be approximately 1.875 meters if the flower bed has dimensions 2m by 6m, or approximately 2.143 meters if the flower bed has dimensions 3m by 4m or 4m by 3m.

Please note that the exact width of the pathway may vary slightly depending on the specific dimensions of the flower bed chosen.

Let me know if you need any further assistance!