What is the maximum height this rock reaches? When will the rock land in the lake?

Question

A hiker throws a rock from the top of a small cliff overlooking a lake. The height of the rock

(in feet) t seconds after it is thrown is given by the formula h(t) = −16t

2 + 64t + 80

What is the maximum height this rock reaches?

When will the rock land in the lake?

Answer 1

A couple important points to remember for problems like this:

1. Whenever we have a squared variable, we know we're graphing a parabola (the "U" shape)
2. The "maximum" or "minimum" of a parabola is also called the "vertex" and we have a formula for finding that
3. The constant term, or the term with no variables, represents the initial height. Since the function you have has a constant term of 80, that means the rock is starting at some height. This hints that the top of the cliff is not considered a "0 height". Logically, I would assume the lake is the 0 height. This means, when the rock hits the lake, h(t) = 0

Given these details, we can start to solve these questions. Finding the vertex can be done in multiple ways, but I recommend the vertex formula (normally, this formula has an x, but I changed it to a "t" to match the given function):

Vertex Formula:

t = -b/(2a)

Standard Form Quadratic:

f(x) = at²+bt+c

Given Function:

h(t) = -16t² + 64t + 80

So, the t-value of our vertex will be:

t = -(64)/(2[-16]) = 2

Now we know that the maximum height will be achieved after 2 seconds. To get the height that will be reached, we just need to plug that 2 in and get our "h".

For when the rock hits the lake, remember that we're looking at a height of 0. This means we'll change "h(t)" to 0, then solve for t. At this point, it's just a regular old quadratic that we can solve with factoring, completing the square, or quadratic formula.

Keep in mind that you'll get two answers when you solve this - one positive number and one negative number. It doesn't make sense that the rock will hit the lake after negative seconds, so you'll throw that answer out and keep just the positive number.

Math sometimes gives us extra information because it thinks the rock has been following this parabola since before it was thrown, which we know isn't true.