Answer:
As per the statement:
The path that the object takes as it falls to the ground can be modeled by
h =-16t^2 + 80t + 300
where
h is the height of the objects and
t is the time (in seconds)
At t = 0 , h = 300 ft
When the objects hit the ground, h = 0
then;
-16t^2+80t+300=0
For a quadratic equation: ax^2+bx+c=0 ......[1]
the solution for the equation is given by:
[tex]x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
On comparing the given equation with [1] we have;
a = -16 ,b = 80 and c = 300
then;
[tex]t= \frac{-80\pm \sqrt{(80)^2-4(-16)(300)}}{2(-16)}[/tex]
[tex]t= \frac{-80\pm \sqrt{6400+19200}}{-32}[/tex]
[tex]t= \frac{-80\pm \sqrt{25600}}{-32}[/tex]
Simplify:
[tex]t = -\frac{5}{2}[/tex] = -2.5 sec and [tex]t = \frac{15}{2}[/tex] = 7.5 sec
Time can't be in negative;
therefore, the time it took the object to hit the ground is 7.5 sec
Answer:
It takes 7.5 seconds the object to hit the ground.
Step-by-step explanation:
An object is propelled upward from the top of a 300 foot building. The path that the object takes as it falls to the ground can be modeled by:
h = -16t^2 + 80t + 300
where:
t is the time (in seconds) and
h is the corresponding height (in feet) of the object.
How long does it take the object to hit the ground?
When the objet hit the ground:
h=0
Then, equaling h (the equation) to zero:
[tex]-16t^{2}+80t+300=0[/tex]
This is a quadratic equation. Using the quadratic formula:
[tex]at^2+bt+c=0; a=-16, b=80, c=300[/tex]
[tex]t=\frac{-b+-\sqrt{b^2-4ac} }{2a}\\ t=\frac{-80+-\sqrt{80^2-4(-16)(300)} }{2(-16)}\\ t=\frac{-80+-\sqrt{6,400+19,200} }{-32}\\ t=\frac{-80+-\sqrt{25,600} }{-32}\\ t=\frac{-80+-160}{-32}[/tex]
Two solutions:
[tex]\left \{ {{t_{1} =\frac{-80+160}{-32} } \atop {t_{2} =\frac{-80-160}{-32} }} \right.[/tex]
[tex]\left \{ {{t_{1} =\frac{80}{-32} } \atop {t_{2} =\frac{-240}{-32} }} \right.[/tex]
[tex]\left \{ {{t_{1} =-2.5} \atop {t_{2} =7.5}} \right.[/tex]
The first solution is not possible, because the time can't be a negative number, then the solution is the second one: t=7.5 seconds
Answer: It takes 7.5 seconds the object to hit the ground.
