The length of a rectangular flower garden is 6 more feet than its width. A walkway 3 feet wide surrounds the outside of the garden. The total area of the walkway itself is 288 square feet. Find the dimensions of the garden.
Let's denote the width of the rectangular flower garden as 'x' feet.
According to the given information, the length of the garden is 6 more feet than its width, so the length can be expressed as (x+6) feet.
To find the dimensions of the garden, we need to subtract the walkway area from the total area of the garden including the walkway.
The area of the garden including the walkway can be calculated by adding twice the width of the garden to the length of the garden.
Total area of the garden including the walkway = (2x + 3)(x+6)
Now, we can set up an equation to solve for 'x':
(2x + 3)(x + 6) - x(x + 6) = 288
Expanding the equation:
(2x^2 + 15x + 18) - (x^2 + 6x) = 288
Simplifying the equation:
x^2 + 9x + 18 = 288
Moving all terms to one side:
x^2 + 9x + 18 - 288 = 0
x^2 + 9x - 270 = 0
We can solve this equation by factoring or by using the quadratic formula. Factoring is not possible in this case, so we will use the quadratic formula to find the value of 'x':
x = (-b ± √(b^2 - 4ac))/(2a)
For the equation x^2 + 9x - 270 = 0, a = 1, b = 9, and c = -270.
x = (-9 ± √(9^2 - 4(1)(-270)))/(2(1))
x = (-9 ± √(81 + 1080))/2
x = (-9 ± √1161)/2
The value of x will be the positive solution, let's approximate it:
x ≈ 12.29 feet
Therefore, the width of the rectangular flower garden is approximately 12.29 feet.
To find the length, we can substitute the value of x into the expression for the length:
Length = x + 6
Length ≈ 12.29 + 6
Length ≈ 18.29 feet
Therefore, the dimensions of the garden are approximately 12.29 feet by 18.29 feet.
To find the dimensions of the garden, we can follow these steps:
Step 1: Let's assign variables to the unknowns in the problem. Let's say the width of the garden is 'x' feet. Since the length is 6 more feet than the width, we can represent the length as 'x + 6' feet.
Step 2: Now, let's calculate the total area of the garden. The area of a rectangle is given by the formula: Area = Length × Width. In this case, the area of the garden is x(x + 6) square feet.
Step 3: Since there is a walkway 3 feet wide surrounding the garden, the width of the garden including the walkway is 'x + 2(3)' feet, which simplifies to 'x + 6' feet. Similarly, the length of the garden including the walkway is 'x + 6 + 2(3)' feet, which simplifies to 'x + 12' feet.
Step 4: The area of the garden including the walkway can be calculated using the formula: Area = Length × Width. In this case, the area is (x + 6)(x + 12) square feet.
Step 5: We are given that the total area of the walkway is 288 square feet. Since the walkway surrounds the garden, the total area of the garden and the walkway combined minus the area of the garden itself will give us the area of the walkway. So, (x + 6)(x + 12) - x(x + 6) = 288.
Step 6: Simplify the equation obtained in Step 5:
(x + 6)(x + 12) - x(x + 6) = 288
(x^2 + 18x + 72) - (x^2 + 6x) = 288
x^2 + 18x + 72 - x^2 - 6x = 288
12x + 72 = 288
Step 7: Solve the equation from Step 6 for the value of 'x'.
12x + 72 = 288
12x = 216
x = 18
Step 8: Now that we have the value of 'x', we can find the dimensions of the garden. The width is x = 18 feet, and the length is x + 6 = 18 + 6 = 24 feet.
Therefore, the dimensions of the garden are 18 feet by 24 feet.
A = wL
x = width of garden
x + 6 = length of garden
Area of garden = x(x + 6)
x + 6 = width of garden and walkway
x + 6 + 6 = x + 12 = length of garden and walkway
Area of garden and walkway =
(x + 6)(x + 12)
Area of garden and walkway - 288 = area of garden
(x + 6)(x + 12) - 288 = x(x + 6)
Solve for x, width of garden
x + 6 = length of garden