Josh kicks a soccer ball with an initial velocity of 18 feet per second out of a window from a height of 12 feet. The function h(t)= -16t^2+18t+12 represents the height of the soccer ball after t seconds.What is the domain of this function in the context of this problem?
the domain of any polynomial is all real numbers, but in this context, I'd have to say it'd be all reals greater than zero.
Getting even more restrictive, we'd need the height to never be negative, so that would restrict the domain to the interval between 0 and where h(t) = 0.
Find that value for t, and that interval will be the domain.
0 greater or equal to t greater or equal to 1.6
To determine the domain of the function in the context of this problem, we need to consider the possible values that t can take.
In this case, t represents time, which cannot be negative in this context because it is measuring how long the soccer ball has been in the air. Therefore, the domain of the function h(t) is only valid for non-negative values of t.
To find the valid values of t, we need to solve for t in the equation h(t) = -16t^2 + 18t + 12 ≥ 0.
First, let's factorize the equation:
-16t^2 + 18t + 12 = -2(8t^2 - 9t - 6)
Next, we can try to factor the quadratic expression inside the parentheses:
-2(8t^2 - 9t - 6) = -2(4t + 3)(2t - 2)
Setting each factor equal to zero and solving for t gives us:
4t + 3 = 0 => t = -3/4
2t - 2 = 0 => t = 1
The equation h(t) is non-negative when 4t + 3 ≤ 0 or 2t - 2 ≥ 0.
From the inequality 4t + 3 ≤ 0, we can determine that t ≤ -3/4. However, this is not possible in this context because time cannot be negative.
From the inequality 2t - 2 ≥ 0, we can determine that t ≥ 1.
Therefore, the valid values of t, or the domain of the function h(t) in this context, are t ≥ 1.