A load of soil that can cover a garden with an area of 1,100 square feet was donated to a community garden. The gardeners would like to create a rectangular garden that is 10 feet more than twice its width.
The equation (?) could be used to find the dimensions of the garden. The perimeter of the garden is approximately (?)
Thank you!
To find the dimensions of the rectangular garden, we can set up an equation.
Let's assume the width of the garden is "w" feet.
According to the given information, the length of the garden is "10 feet more than twice its width." So, the length would be (2w + 10) feet.
Since the garden area is 1,100 square feet, the equation to express this is:
(w) * (2w + 10) = 1100
Now, let's solve this equation to find the value of "w."
w(2w + 10) = 1100
2w^2 + 10w - 1100 = 0
To solve this quadratic equation, we can try factoring or using the quadratic formula. In this case, let's use factoring.
2w^2 + 10w - 1100 = (2w - 40)(w + 55) = 0
Setting each factor to zero gives us:
2w - 40 = 0 or w + 55 = 0
From the first equation, we find w = 20. From the second equation, we find w = -55. Since the width cannot be negative, we discard the value of -55.
Therefore, the width of the rectangular garden is 20 feet.
To find the length, we substitute the value of w back into the equation (2w + 10):
Length = 2w + 10 = 2(20) + 10 = 40 + 10 = 50 feet.
So, the dimensions of the garden are 20 feet (width) and 50 feet (length).
To find the perimeter of the garden, we can use the formula:
Perimeter = 2(length + width)
Plugging in the values, we get:
Perimeter = 2(50 + 20) = 2(70) = 140 feet.
Therefore, the perimeter of the garden is approximately 140 feet.
To find the dimensions of the garden, we can set up an equation related to the area of the garden.
Let's assume the width of the rectangular garden is "x" feet. According to the given information, the length of the garden is stated as "10 feet more than twice its width," which translates to (2x + 10) feet.
The area of a rectangle can be calculated by multiplying its length and width. In this case, the equation is:
Area = Length × Width
1100 = (2x + 10) × x
To find the approximate perimeter of the garden, we need to know the dimensions (length and width). However, the original equation only provides us with the area.
To proceed, we need to solve the equation for "x". We can start by expanding:
1100 = 2x² + 10x
Rearranging the equation to make it quadratic:
2x² + 10x - 1100 = 0
Now, we can solve the equation for "x" using factoring, completing the square, or the quadratic formula. In this case, let's use factoring. Factoring out a common factor of 2:
2(x² + 5x - 550) = 0
Next, we can try to factor the expression inside the parentheses. To find suitable factors of -550, we need to find two numbers that multiply to -550 and add up to +5. After exploring the options, we find that +25 and -22 satisfy these conditions:
2(x + 25)(x - 22) = 0
Now we have two possible solutions:
1) x + 25 = 0 => x = -25 (not a valid solution since the width cannot be negative)
2) x - 22 = 0 => x = 22
The width cannot be negative, so the valid width of the garden is 22 feet.
To find the length, we can substitute this value back into the equation:
Length = 2x + 10
Length = 2(22) + 10
Length = 44 + 10
Length = 54 feet
Now we have the dimensions of the garden. The width is 22 feet, and the length is 54 feet.
To calculate the approximate perimeter, we use the formula:
Perimeter = 2 × (Width + Length)
Perimeter = 2 × (22 + 54)
Perimeter = 2 × 76
Perimeter = 152 feet
Therefore, the perimeter of the garden is approximately 152 feet.
Thank you very much!
L * W = 1100
L = 2 W + 10
substituting ... W (2 W + 10) = 1100 ... W^2 + 5 W - 550 = 0
use the quadratic formula to find W
... substitute back to find L
perimeter = 2 (L + W)