the height of a soccer goal is 0.5 feet more than one third of its width. if the area of this goal is 117ft^2, find the width of the goal
Let x = width, then 1/3x+.5 = height
Area = h * w
To find the width of the soccer goal, we first need to set up an equation based on the given information. Let's assume the width of the goal is represented by 'x' feet.
According to the problem, the height of the goal is 0.5 feet more than one third of its width. So, the height of the goal can be expressed as (1/3)x + 0.5 feet.
The area of the goal can be calculated by multiplying the width and height:
Area = Width × Height
117 ft² = x × [(1/3)x + 0.5]
Now we can solve this equation to find the width of the goal.
To do that, let's first simplify the equation:
(1/3)x² + 0.5x = 117
Multiply both sides of the equation by 3 to eliminate the fraction:
x² + 1.5x = 351
Rearrange the equation to form a quadratic equation:
x² + 1.5x - 351 = 0
Now we have a quadratic equation in the form of ax² + bx + c = 0, where a = 1, b = 1.5, and c = -351.
To solve this quadratic equation, we can use either factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Plugging in the values:
x = (-1.5 ± √(1.5² - 4(1)(-351))) / 2(1)
Simplifying further:
x = (-1.5 ± √(2.25 + 1404)) / 2
x = (-1.5 ± √(1406.25)) / 2
x = (-1.5 ± 37.5) / 2
This gives us two possible values for the width of the goal:
x₁ = (-1.5 + 37.5) / 2 = 36 / 2 = 18 feet
x₂ = (-1.5 - 37.5) / 2 = -39 / 2 = -19.5 feet
Since the width of the goal cannot be negative, the width of the goal is 18 feet.