The dimensions of a playground are 50 feet by 75 feet. You want
to double its area by adding the same distance x to the length and
width. Find x and the new dimensions of the playground.
distance to be added = x
original area = 50x75 = 3750 ft^2
double that area = 7500 ft^2
(50+x)(75+x) = 7500
3750 + 125x + x^2 = 7500
x^2 + 125x - 3750 = 0
(x+150)(x-25) = 0
x = 25 or x is a negative
25 feet should be added to each dimension
check:
new length = 100
new width = 75
new area = 7500 , which is twice the original,
all is good
To double the area of the playground, we need to add the same distance x to both the length and the width.
The original area of the playground is given by the product of the length and the width: 50 ft * 75 ft = 3750 sq ft.
To find x, we can set up the equation:
(50 ft + x) * (75 ft + x) = 2 * 3750 sq ft.
Simplifying the equation, we get:
3750 + 125x + 75x + x^2 = 7500.
Combining like terms, we have:
x^2 + 200x + 3750 - 7500 = 0.
Simplifying further, we get:
x^2 + 200x - 3750 = 0.
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a.
For this equation, a = 1, b = 200, and c = -3750. Plugging in these values, we get:
x = (-200 ± sqrt(200^2 - 4 * 1 * -3750)) / (2 * 1).
Simplifying, we have:
x = (-200 ± sqrt(40000 + 15000))/ 2.
x = (-200 ± sqrt(55000))/ 2.
Taking the positive square root, we get:
x ≈ (-200 + sqrt(55000))/ 2.
Using a calculator, we find that x ≈ 16.16 feet (rounded to two decimal places).
To find the new dimensions of the playground, we add this distance to both the length and the width.
The new length = 50 ft + x ≈ 50 ft + 16.16 ft ≈ 66.16 ft.
The new width = 75 ft + x ≈ 75 ft + 16.16 ft ≈ 91.16 ft.
Therefore, the new dimensions of the playground are approximately 66.16 feet by 91.16 feet.
To double the area of the playground, we need to find the value of x and the new dimensions of the playground.
The current dimensions of the playground are given as 50 feet by 75 feet. Let's calculate the current area of the playground:
Area = Length × Width
Area = 50 ft × 75 ft
Area = 3750 square feet
To double the area, we need to add the same distance x to both the length and the width. Let's assume x is the distance we need to add.
The new length would be 50 ft + x, and the new width would be 75 ft + x.
The new area of the playground would be:
New Area = (50 ft + x) × (75 ft + x)
We want the new area to be double the current area, so:
2 × Current Area = New Area
2 × 3750 square feet = (50 ft + x) × (75 ft + x)
Now, let's solve this equation to find the value of x and the new dimensions of the playground:
2 × 3750 = (50 + x) × (75 + x)
7500 = 3750 + 125x + 50x + x^2
Rearranging the equation, we have:
0 = x^2 + 175x - 3750
Now, we can solve this quadratic equation for x. By factoring or using the quadratic formula, we find that x can be x = 25 or x = -150.
Since we cannot have a negative distance, we ignore x = -150, and we have x = 25 as the valid value.
Therefore, the distance x is 25 feet, and the new dimensions of the playground would be:
New Length = 50 ft + x = 50 ft + 25 ft = 75 ft
New Width = 75 ft + x = 75 ft + 25 ft = 100 ft
Hence, the new dimensions of the playground would be 75 feet by 100 feet.