a rectangular pool has an area of 880ft. the lenght is 10 feet longer than thw width. find the dimensions of the pool. solve by completing the square. round answer to the nearest tenth of a foot.
X*(x+10)=880ft
X square=870 ft u find the square root of 870 and plug it in to the first equation
About 25 feet and 35 feet (when rounded)
Let's solve this problem step by step using the completing the square method.
Step 1: Let's assume the width of the pool as "x" feet.
Step 2: According to the given information, the length of the pool is 10 feet longer than the width. So, the length would be "x + 10" feet.
Step 3: The area of a rectangle is given by the formula: Area = Length * Width.
So, we can write the equation as: x(x + 10) = 880.
Step 4: Expand the equation:
x^2 + 10x = 880
Step 5: To complete the square, add the square of half the coefficient of 'x' on both sides of the equation:
x^2 + 10x + (10/2)^2 = 880 + (10/2)^2
x^2 + 10x + 25 = 880 + 25
x^2 + 10x + 25 = 905
Step 6: Rewrite the left side of the equation as a perfect square:
(x + 5)^2 = 905
Step 7: Take the square root on both sides to solve for 'x':
√(x + 5)^2 = √905
(x + 5) = ± √905
Step 8: Solve for 'x':
x + 5 = ± √905
x = -5 ± √905
Since we're dealing with physical dimensions, the length and width cannot be negative. Therefore, we can take the positive value of x.
Step 9: Calculate the value of 'x':
x = -5 + √905 ≈ 28.1 feet
(The width of the pool)
Step 10: Substitute the value of 'x' in the equation for the length:
Length = x + 10
Length = 28.1 + 10 ≈ 38.1 feet
So, the dimensions of the pool are approximately 28.1 feet by 38.1 feet.
To solve this problem using completing the square method, we'll start by setting up the equation.
Let's assume that the width of the pool is represented by "x" feet.
According to the problem, the length is 10 feet longer than the width, so the length would be (x + 10) feet.
We know that the area of a rectangle is given by the formula: Area = Length × Width.
In this case, the area of the rectangular pool is given as 880ft². So we can set up the equation:
(x + 10) × x = 880
Now, let's simplify this equation:
x² + 10x = 880
To complete the square, we need to add a constant to both sides of the equation. The constant we add is half the coefficient of x squared term, which is (10/2)² = 25.
Adding 25 to both sides of the equation, we get:
x² + 10x + 25 = 880 + 25
Now, let's simplify further:
(x + 5)² = 905
To solve for x, we can now take the square root of both sides of the equation:
√(x + 5)² = √905
x + 5 = ±√905
Subtracting 5 from both sides, we get:
x = -5 ± √905
Since we cannot have a negative width for a pool, we'll disregard the solution with the negative value.
So, x = -5 + √905
To find the dimensions of the pool, we can substitute this value of x back into our expression for the length:
Length = x + 10
Length = (-5 + √905) + 10
Calculating this expression, we find:
Length ≈ 8.8 feet
Therefore, the dimensions of the pool are approximately:
Width ≈ -5 + √905 feet
Length ≈ 8.8 feet