Question

An object is thrown straight upward from the ground with an initial speed of 30 feet per second. Its height (h) after t seconds is given by the equation h(t) = -5t2 + 30t. Find the domain for h(t).
A. (-∞, 0)
B. (6, ∞)
C. (0, 6)
D. All real numbers

Answer 1

(0,6)

Answer 2

Do u want explanation?

Answer 3

We know that t represents time and h(t) represents the height of the ball. In real life, we know we can't have negative height and neither can we have negative time. Thus, the domain must be those values of time that are positive and for which positive values of h(t) exist.

I will first factor h(t) so it's easier to get information out of it. Factoring gives:

h(t) = -5t^2 + 30t = -5t(t - 6) --> x-intercepts: t = 0, 6

We know that this will be a parabola when graphed. We also know that the leading coefficient is negative so it will be reflected in the x-axis. But this equation also has "+ 30t" in it. This means that there is a vertical shift on the parabola and we shift it upwards. There part of the parabola which lies above the axis will then be between the zeroes (at t = 0, 6) which will be the part where height is positive, and that's what we want. We also know that x-intercepts at t = 0 and t = 6 also means that time is non-negative, which is sufficient.

This must mean that any part of the parabola lying outside of this interval of [0, 6], will be below the x-axis since the parabola is reflected in the x-axis, and we don't want those negative height values. So the only place we find non-negative h(t) values for non-negative values of t is in the interval [0, 6].

Therefore, the domain must be [0, 6].

@katlin95