This is a classic type of question. The function gives the height h at a given time t after the throw. In this case, we are given the height and we will be solving for the time. The rock is at 16 feet on its way up and on its way down. From the sound of the question, they want the time when the rock first gets to 16 feet from the surface of Mars.
h: height in feet from the Mars surface
t: time in seconds from when thrown
h(t) = –12t2 + 24t + 6
Simply fill in 16 for h(t) to start.
16 = –12t2 + 24t + 6
Now get all the terms on the same side of the equation, leaving 0 on the other side.
16 – 16 = –12t2 + 24t + 6 – 16
0 = –12t2 + 24t –10
I'm going to multiply the entire equation by (–1) to get:
0 = 12t2 – 24t + 10
Now I'm going to divide the entire equation by 2 to simplify it:
0 = 6t2 –12t + 5
Since I cannot figure out how to factor this with integers, I'm going to use the quadratic formula:
–(–12) ± /√[(–12)2 – (4)(6)(5)]
t = _______________________
2(6)
12 ± √[144 –120]
t = ______________________
12√
12 ± √24
t = _____________________
12
12 ± √4√6
t = ___________
12
12 ± √4√6
t = ___________
12
12 ± 2√6
t = ___________
12
That is the precise amount of time in seconds WHEN YOU USE THE SUBTRACTION symbol.
Since we are asked to round the time to the nearest hundredth of a second, we will convert √6 to an approximately equal decimal number, 2.4495.
12 ± (2)(2.4495)
t ≈ ________________
12
12 ± 4.899
t ≈ ________________
12
t ≈ 0.59 seconds
or
t ≈ 1.41 seconds
So, at approximately 0.59 seconds after being thrown, the rock will be 16 feet above the Mars surface. After 1.14 seconds, the rock again will be 16 feet above the Mars surface on its way down to Mars.
