You are planning a vegetable garden and you lay out a rectangular design 10 feet wide by 20 feet long. After laying out the design you decide you want a larger garden and decide to increase the length of the garden by a length of 2x feet and increase the width by a length of x feet. You have enough dirt to cover an area of 600 square feet, and you want to make the garden as large as possible. What are the dimensions of the finished garden? Round your answer to the nearest foot.
600=(2x+20)(x+10)
600=2x^2 +40x+200
x^2+20x-500=0
x=(-20+-sqrt(400+2000) )/2
x= -10 +-sqrt(600)
x=sqrt(600) -10
Let's assume the increase in length is represented by "2x" feet and the increase in width is represented by "x" feet.
The current dimensions of the garden are:
Width = 10 feet
Length = 20 feet
After increasing the length and width, the new dimensions of the garden would be:
Width = 10 + x feet
Length = 20 + 2x feet
The area of the garden is given as 600 square feet, so we can make the following equation using the area formula for a rectangle:
(10 + x) * (20 + 2x) = 600
Let's solve this equation to find the values of "x" and determine the dimensions of the finished garden.
Expanding the equation, we get:
200 + 40x + 10x + 2x^2 = 600
Rearranging the terms:
2x^2 + 50x - 400 = 0
Now we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, the values are:
a = 2, b = 50, c = -400
Substituting the values into the quadratic formula:
x = (-50 ± √(50^2 - 4*2*(-400))) / (2*2)
x = (-50 ± √(2500 + 3200)) / 4
x = (-50 ± √5700) / 4
Calculating the square root:
x = (-50 ± 75.5) / 4
Now, consider both cases separately:
Case 1:
x = (-50 + 75.5) / 4
x = 25.5 / 4
x ≈ 6.375
Case 2:
x = (-50 - 75.5) / 4
x = -125.5 / 4
x ≈ -31.375
Since negative values for the dimensions do not make sense in this context, we'll disregard the second case.
Therefore, the value of 'x' is approximately 6.375 feet.
Now, let's find the dimensions of the finished garden:
Width = 10 + x ≈ 10 + 6.375 ≈ 16.375 feet (rounded to the nearest foot)
Length = 20 + 2x ≈ 20 + 2(6.375) ≈ 33.75 feet (rounded to the nearest foot)
So, the dimensions of the finished garden would be approximately 16 feet wide by 34 feet long.
To find the dimensions of the finished garden, we need to maximize the area of the garden while considering the given constraints.
Let's assume the increase in length is 2x feet and the increase in width is x feet. Therefore, the new length of the garden will be 20 + 2x, and the new width will be 10 + x.
The formula for the area of a rectangle is length multiplied by width. So, the area of the new garden will be:
Area = (20 + 2x) * (10 + x)
We know that the area of the new garden should be less than or equal to 600 square feet, so we can set up the following inequality:
(20 + 2x) * (10 + x) ≤ 600
Now, we can solve this inequality to find the maximum dimensions of the finished garden.
Expanding the inequality:
(20 + 2x) * (10 + x) ≤ 600
200 + 40x + 20x + 4x^2 ≤ 600
4x^2 + 60x + 200 ≤ 600
4x^2 + 60x - 400 ≤ 0
Simplifying the equation further:
x^2 + 15x - 100 ≤ 0
Now, we need to find the values of x that satisfy this inequality. We can do this by factoring or using the quadratic formula. Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 1, b = 15, and c = -100.
x = (-15 ± √(15^2 - 4(1)(-100))) / 2(1)
x = (-15 ± √(225 + 400)) / 2
x = (-15 ± √625) / 2
x = (-15 ± 25) / 2
So, the possible values of x are x = 5 or x = -10. Since we are looking for positive lengths, we can disregard the negative value of x.
Therefore, x = 5.
Now, we can find the dimensions of the finished garden:
New length = 20 + 2x = 20 + 2(5) = 20 + 10 = 30 feet
New width = 10 + x = 10 + 5 = 15 feet
So, the dimensions of the finished garden will be approximately 30 feet by 15 feet.