Allison has a square root flower garden and Lauren has a rectangular flower garden the width of the rectangular garden is 2 yards less than a side of the squaare garden and the length of the rectangular garden is 5 yards more than a side of the square garden if the sum of the areas of both Allison and Lauren gardens is 200 square yards find the measure of a side of the square garden. show workout please
Well, well, well, let's put our clown shoes on and solve this flower garden puzzle!
Let's call the side length of Allison's square garden "x" yards. Therefore, the area of her garden is x^2 square yards. Stick with me, we're just warming up!
Now let's move on to Lauren's rectangular garden. The width is 2 yards less than a side of Allison's square garden, so the width is (x - 2) yards. The length is 5 yards more than a side of the square garden, so the length is (x + 5) yards. My oh my, we're almost there!
The area of Lauren's rectangular garden is the product of its length and width, which is (x - 2)(x + 5) square yards.
Now, according to our problem, the sum of the areas of both gardens is 200 square yards. So, we set up an equation:
x^2 + (x - 2)(x + 5) = 200
Are you ready? Let's simplify this equation and solve for x:
x^2 + (x^2 + 5x - 2x - 10) = 200
2x^2 + 3x - 10 = 200
2x^2 + 3x - 210 = 0
Now, let's use the quadratic formula to find the value of x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from our equation:
x = (-3 ± √(3^2 - 4*2*(-210))) / (2*2)
x = (-3 ± √(9 + 3360)) / 4
x = (-3 ± √3369) / 4
Haha, sorry, but the square root of 3369 is not a pretty number. It's approximately 58.05 (rounded to two decimal places).
So, the side length of Allison's square garden, approximately, is 58.05 yards. Ta-da! I hope you had as much fun as I did solving this garden circus!
Let's represent the side of the square garden as "x" yards.
The width of the rectangular garden is 2 yards less than the side of the square garden, so it would be (x - 2) yards.
The length of the rectangular garden is 5 yards more than a side of the square garden, so it would be (x + 5) yards.
To find the area of the square garden, we use the formula A = side². So, the area of the square garden is x² square yards.
To find the area of the rectangular garden, we use the formula A = length × width. So, the area of the rectangular garden is (x + 5)(x - 2) square yards.
Given that the sum of the areas of both gardens is 200 square yards, we can write the equation:
x² + (x + 5)(x - 2) = 200
Expanding and simplifying:
x² + (x² + 5x - 2x - 10) = 200
x² + x² + 3x - 10 = 200
Combining like terms:
2x² + 3x - 10 - 200 = 0
2x² + 3x - 210 = 0
Now, we need to solve this quadratic equation for x.
By factoring or using the quadratic formula, we find that x≈11.16 or x≈-9.465.
Since side lengths can't be negative, we discard the negative value.
Therefore, the measure of a side of the square garden is approximately 11.16 yards.
To solve this problem, let's assign variables and set up equations to represent the given information.
Let:
x = the side length (in yards) of the square garden
w = the width (in yards) of the rectangular garden
l = the length (in yards) of the rectangular garden
From the given information, we can form the following equations:
1. The area of Allison's square garden (A1):
A1 = x * x = x^2
2. The area of Lauren's rectangular garden (A2):
A2 = w * l
According to the problem statement, the width of the rectangular garden is 2 yards less than a side of the square garden:
w = x - 2
The length of the rectangular garden is 5 yards more than a side of the square garden:
l = x + 5
The sum of the areas of both gardens is 200 square yards:
A1 + A2 = 200
Substituting the values of A1, A2, w, and l into the equation, we have:
x^2 + (x - 2)(x + 5) = 200
Expanding and simplifying the equation:
x^2 + (x^2 + 3x - 10 ) = 200
2x^2 + 3x - 10 = 200
2x^2 + 3x - 210 = 0
Now, we have a quadratic equation. To solve it, we can either factor, complete the square, or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 2, b = 3, and c = -210. Plugging these values into the formula:
x = (-3 ± √(3^2 - 4 * 2 * -210)) / (2 * 2)
x = (-3 ± √(9 + 1680)) / 4
x = (-3 ± √1689) / 4
Since we are discussing measurements, we will only consider the positive value for x:
x = (-3 + √1689) / 4 ≈ 10.85
Therefore, the measure of a side of the square garden is approximately 10.85 yards.