A rectangular garden has dimensions 3m by 4m. A path is built around the garden. The area of the garden and path is 6 times as a great as the area of the garden. What is the width of the path?
(3 + 2w) * (4 + 2w) = 3 * 4 * 6
4 w^2 + 14 w + 12 = 72 ... 2 w^2 + 7 w - 30 = 0
solve for w
Ag = Area of the garden.
Ap = Area of the path.
Ag + Ap = 6Ag.
3*4 + Ap = 6(3*4),
Ap = 60 sq. meters.
L*W = 60.
L/W = 4/3,
L = 4W/3,
4W/3 * W = 60,
4W^2 = 180,
W = ?
Let's denote the width of the path as "x".
The area of the rectangular garden is given by the formula: length * width = 3m * 4m = 12m^2.
Since the path is built around the garden, the dimensions of the garden and path together would be:
(3m + 2x) * (4m + 2x).
The area of the garden and path is given as 6 times the area of the garden, so we have the following equation:
(3m + 2x) * (4m + 2x) = 6 * 12m^2.
Expanding the equation and simplifying, we get:
12m^2 + 6x(3m + 4m) + 4x^2 = 6 * 12m^2.
Simplifying further, we have:
12m^2 + 6x(7m) + 4x^2 = 6 * 12m^2.
Next, we can cancel out the common term "12m^2" from both sides:
6x(7m) + 4x^2 = 6 * 12m^2 - 12m^2.
Now, simplify the right side:
6x(7m) + 4x^2 = 72m^2.
Divide both sides by 2 to simplify:
3x(7m) + 2x^2 = 36m^2.
Finally, rearrange the equation to get it into standard form:
2x^2 + 21xm - 36m^2 = 0.
Now we can solve this quadratic equation to find the values of "x", which will give us the width of the path.
To find the width of the path, we need to determine the area of the garden and the area of the garden and path combined. Let's first calculate the area of the garden.
The area of the garden is given by multiplying its length and width:
Area of the garden = Length × Width
Area of the garden = 3m × 4m
Area of the garden = 12m²
Next, we need to find the area of the garden and path combined. According to the problem, this area is 6 times as great as the area of the garden:
Area of garden and path = 6 × Area of the garden
Area of garden and path = 6 × 12m²
Area of garden and path = 72m²
Now, we can find the combined area of the garden and path by adding the width of the path to the length and width of the garden.
Let's assume the width of the path is 'x'. We can express the length and width of the garden and path combined as:
Length = Length of garden + 2 × Width of path
Width = Width of garden + 2 × Width of path
Plugging in the values we have:
Length = 3m + 2x
Width = 4m + 2x
Now, we can calculate the area of the garden and path combined by multiplying the length and width:
Area of garden and path = Length × Width
72m² = (3m + 2x) × (4m + 2x)
To solve for 'x', we can expand the equation, set it equal to zero, and solve for 'x' using factoring or the quadratic formula:
72m² = 12m² + 20xm + 8xm + 4x²
0 = 4x² + 28xm + 12m² - 72m²
0 = 4x² + 28xm - 60m²
Now, we can simplify the equation to:
0 = x² + 7xm - 15m²
This equation represents a quadratic equation. We need more information to find the value of 'x' directly.