To determine the instantaneous rate of change of the height of the rock after 4 seconds, we need to find the derivative of the height function [tex]\( h(t) \)[/tex].
The height function given is:
[tex]\[ h(t) = -5t^2 + 10t + 120 \][/tex]
The instantaneous rate of change of the function [tex]\( h(t) \)[/tex] at a specific time [tex]\( t \)[/tex] is given by the derivative of [tex]\( h(t) \)[/tex] with respect to [tex]\( t \)[/tex].
First, let's find the derivative [tex]\( h'(t) \)[/tex]:
[tex]\[ h(t) = -5t^2 + 10t + 120 \][/tex]
Using the power rule for differentiation, where [tex]\(\frac{d}{dt}(t^n) = nt^{n-1}\)[/tex]:
1. The derivative of [tex]\(-5t^2\)[/tex] is [tex]\(-10t\)[/tex].
2. The derivative of [tex]\(10t\)[/tex] is [tex]\(10\)[/tex].
3. The derivative of the constant [tex]\(120\)[/tex] is [tex]\(0\)[/tex].
Thus, the derivative [tex]\( h'(t) \)[/tex] is:
[tex]\[ h'(t) = -10t + 10 \][/tex]
Next, we need to evaluate this derivative at [tex]\( t = 4 \)[/tex] to find the instantaneous rate of change at that specific time:
[tex]\[ h'(4) = -10(4) + 10 \][/tex]
Evaluating this, we get:
[tex]\[ h'(4) = -40 + 10 \][/tex]
[tex]\[ h'(4) = -30 \][/tex]
Therefore, the instantaneous rate of change of the height of the rock after 4 seconds is [tex]\(-30\)[/tex] meters per second.
