In sixteen states and the District of Columbia, the legal blood-alcohol level is 0.08. Use the formula A(x) = -0.0082x2 + 0.06x to compute blood-alcohol levels as a function of x (the time in hours). When is the first time that a person can be arrested for drunken driving after drinking 5 ounces of 80-proof whiskey in those states?
1.6 or 1.8
To find the time at which a person can be arrested for drunken driving after drinking 5 ounces of 80-proof whiskey, we need to find the value of x in the equation A(x) = -0.0082x^2 + 0.06x when A(x) becomes greater than or equal to 0.08.
The equation A(x) = -0.0082x^2 + 0.06x represents the blood-alcohol level as a function of time in hours. The variable x represents the time in hours, and A(x) represents the blood-alcohol level.
To solve the equation, we need to set A(x) greater than or equal to 0.08 and solve for x:
-0.0082x^2 + 0.06x ≥ 0.08
To simplify the equation, we can multiply both sides by -1000 to remove the decimals:
-8.2x^2 + 60x ≥ 80
Rearranging the equation, we have:
8.2x^2 - 60x + 80 ≤ 0
This is a quadratic inequality. To solve it, we can either factor or use the quadratic formula. In this case, the quadratic formula would be more effective.
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation, a = 8.2, b = -60, and c = 80. Plugging these values into the formula, we can find the solutions for x.
x = (-(-60) ± √((-60)^2 - 4(8.2)(80))) / (2(8.2))
x = (60 ± √(3600 - 2624)) / (16.4)
x = (60 ± √976) / 16.4
We have two solutions for x:
x1 = (60 + √976) / 16.4
x2 = (60 - √976) / 16.4
Calculating these values, we get:
x1 ≈ 1.6
x2 ≈ 1.8
Therefore, the first time a person can be arrested for drunken driving after drinking 5 ounces of 80-proof whiskey in those states is approximately 1.6 hours or 1 hour and 36 minutes.