A rock is dropped from a 600 foot tower.
The height of the rock as a function of time
can be modeled by the equation:
h(t) = −16t
2
+ 600. During what period of
time will the height of the rock be greater
than 300 above the ground?
height = -16t^2 + 600
when does it reach a height of 300?
300 = -16t^2 + 600
16t^2 = 300
t^2 = 300/16
t = √300/4
= 10√3/4 = appr 4.33 seconds
It will be above or equal to a height of 300 ft for the first 4.33 seconds
To find the period of time when the height of the rock is greater than 300 above the ground, we need to solve the inequality:
h(t) > 300
Substituting the equation for height, we have:
-16t^2 + 600 > 300
First, let's subtract 300 from both sides of the inequality:
-16t^2 + 300 > 0
Now, let's divide both sides of the inequality by -16. Since we are dividing by a negative number, the inequality sign will flip:
t^2 - 18.75 < 0
To solve this quadratic inequality, we can find the values of t that make the expression equal to zero, and then determine the intervals where the expression is greater or less than zero.
To find the values of t that make the expression equal to zero, we set:
t^2 - 18.75 = 0
Now we can solve for t:
t^2 = 18.75
t = ±√18.75
t ≈ ±4.33
So, the values of t that make the expression t^2 - 18.75 equal to zero are t ≈ 4.33 and t ≈ -4.33.
Now we can determine the intervals where the expression t^2 - 18.75 is greater or less than zero.
To do this, let's create a number line and plot the values of t:
<---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->
-5 -4 -3 -2 -1 0 1 2 3 4 5
We can see that the intervals where the expression t^2 - 18.75 < 0 are approximately t < -4.33 and 4.33 < t.
Therefore, the period of time when the height of the rock is greater than 300 above the ground is approximately t < -4.33 and 4.33 < t.