precalc question: find the domain of this function?

Question

A firecracker is launched straight up, and its height is a function of time, h(t) = −16t² + 128t, where h is the height in feet and t is the time in seconds, with t = 0 corresponding to the instant it launches. What is the height 4 seconds after launch? What is the domain of this function?

Answer 1

To find the height 4 seconds after launch, plug in t = 4 in the formula for h: h(4) = -16(4²) + 128(4) = -256 + 512 = 256 ft. The domain of h is the set of all t values for which h(t) is defined. Since h(t) must be nonnegative, to find the domain we must solve the inequality -16t² + 128t ≥ 0, i.e., -16t(t - 8) ≥ 0. The zeros of -16t(t - 8) are t = 0 and t = 8. We verify the truth of the inequality in the intervals t < 0, 0 < t < 8, and t > 8, using the sign test. For t < 0 we may take t = -1 as a test point: -16(-1)(-1 - 8) = 16(-9) < 0. So the inequality is false for t < 0. For 0 < t < 8, we may take t = 1 as a test point: -16(1)(1 - 8) = -16(-7) > 0. Thus, the inequality is true for 0 < t < 8. Finally, for t > 8, we may take t = 10 as a test point: -16(10)(10 - 8) = -16(10)(2) < 0. Therefore, the inequality is false for t > 8. Since t = 0 and t = 8 also satisfy the inequality, we determine that the solution set is [0, 8]. So the domain of h is the closed interval [0, 8].