Solve by completing the square: A rectangular patio has a length of x + 6 m, a width of x + 8 m, and a total area of 400 m². Find the dimensions to the nearest tenth.
Find the radius of a circular lid with an area of 12in.².
(x+6)(x+8)=400
x^2+14x+48=400
x^2+14x+49=401
(x+7)^2 = 401
x+7 = ±√401
x = -7±√401
πr^2 = 12
r^2 = 12/π
r = √(12/π)
Thank you SO MUCH anonymous
To solve the first problem, we need to complete the square.
The area of a rectangle is given by the formula A = length × width. In this case, the area is 400 m², and the length is x + 6 m, and the width is x + 8 m. So we have the equation:
400 = (x + 6)(x + 8)
Expanding the right side of the equation:
400 = x^2 + 8x + 6x + 48
Combining like terms:
400 = x^2 + 14x + 48
To complete the square, we need to add and subtract the square of half the coefficient of x (in this case, it's 14/2 = 7) to both sides of the equation:
400 + 7^2 = x^2 + 14x + 49 + 48
Simplifying:
400 + 49 = x^2 + 14x + 97
449 = x^2 + 14x + 97
Rearranging the terms:
x^2 + 14x - 352 = 0
To solve for x, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 14, and c = -352. Substituting these values into the equation:
x = (-14 ± √(14^2 - 4(1)(-352))) / (2(1))
Simplifying further:
x = (-14 ± √(196 + 1408)) / 2
x = (-14 ± √1604) / 2
x = (-14 ± 40.05) / 2
Therefore, the possible values for x are:
x = (-14 + 40.05) / 2 ≈ 13.03
x = (-14 - 40.05) / 2 ≈ -27.03
The dimensions of the rectangle are approximately x + 6 = 13.03 + 6 = 19.03 m for the length, and x + 8 = 13.03 + 8 = 21.03 m for the width (rounded to the nearest tenth).
For the second problem, we are given the area of a circular lid, which is 12 in². The formula for the area of a circle is A = πr², where A is the area and r is the radius.
We can rearrange the formula to solve for the radius:
r² = A / π
Substituting the given values:
r² = 12 in² / π
To find the radius, we can take the square root of both sides of the equation:
r = √(12 in² / π)
Using a calculator to evaluate this expression:
r ≈ 1.95 inches (rounded to the nearest tenth)
Therefore, the radius of the circular lid is approximately 1.95 inches.
To solve the first question, we need to complete the square.
Step 1: Start with the equation representing the area of the rectangular patio:
length * width = area
Step 2: Substitute the given expressions for length and width:
(x + 6)(x + 8) = 400
Step 3: Expand the equation:
x^2 + 14x + 48 = 400
Step 4: Move the constant term to the right side of the equation:
x^2 + 14x - 352 = 0
Step 5: To complete the square, we need to add a constant term to both sides of the equation. Take half of the coefficient of the x-term (14) and square it. The result is 7^2 = 49. Add this constant term to both sides of the equation:
x^2 + 14x + 49 - 352 + 49 = 49
Step 6: Simplify the equation:
(x + 7)^2 - 303 = 0
Step 7: Solve for x by taking the square root of both sides:
x + 7 = ±√303
Step 8: Subtract 7 from both sides:
x = -7 ± √303
Approximating to the nearest tenth:
x ≈ -7 + √303 ≈ 13.7 meters
x ≈ -7 - √303 ≈ -20.7 meters
Since lengths cannot be negative in this context, the length of the patio is approximately 13.7 meters and the width is approximately 13.7 + 2 = 15.7 meters.
To solve the second question, we can use the formula for the area of a circle:
Area = π * radius^2
Given Area = 12 in², we can solve for the radius:
12 = π * radius^2
Divide both sides by π:
radius^2 = 12/π
Take the square root of both sides:
radius = √(12/π) ≈ 1.94 inches (rounded to the nearest tenth)
Therefore, the radius of the circular lid is approximately 1.94 inches.