A brick border that is x feet wide is built around a rectangular patio. The patio is 6 feet wide and 8 feet long. The combined area of the patio and the brick border is 120 square feet. What is the width of the brick border?
To find the width of the brick border, we need to subtract the area of the patio from the combined area of the patio and the border.
Let's start by finding the area of the patio. The formula for the area of a rectangle is length multiplied by width, so the area of the patio is 6 feet x 8 feet = 48 square feet.
Next, we subtract the area of the patio from the combined area of the patio and the border to find the area of the brick border. The combined area is given as 120 square feet, so the area of the border is 120 square feet - 48 square feet = 72 square feet.
Finally, since we are given that the width of the border is x feet, we can set up an equation to solve for x using the area of the border. The area of a rectangle is length multiplied by width, so x feet multiplied by the length of the border is equal to the area of the border, which is 72 square feet.
Therefore, x feet x (6 + 2x) feet = 72 square feet.
To solve this equation, we can simplify and rewrite it as a quadratic equation:
2x^2 + 6x - 72 = 0
Now, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the value of x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 2, b = 6, and c = -72. Substituting these values into the formula, we have:
x = (-6 ± √(6^2 - 4(2)(-72))) / (2(2))
= (-6 ± √(36 + 576)) / 4
= (-6 ± √612) / 4
To simplify further, we can factor out 36 from √612:
x = (-6 ± √(36(1 + 17))) / 4
= (-6 ± 6√(1 + 17)) / 4
= -3 ± 3√(1 + 17) / 2
= -3 ± 3√18 / 2
= -3 ± 3√(9 x 2) / 2
= -3 ± 3√9√2 / 2
= -3 ± 3 x 3√2 / 2
= -3 ± 9√2 / 2
So, the possible values for x are -3 + 9√2 / 2 and -3 - 9√2 / 2.
Since we are looking for the width, x must be a positive value. Therefore, the width of the brick border is -3 + 9√2 / 2 feet.