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In this problem I'm given the height of the rocket in feet
t seconds after it is launched from forty feet above the ground
and this is given by the equation h(t) is equal to a negative 16t squared is equal to a negative sixteen t squared
plus 80t plus 40
and i wanna know how long it will take the rocket to hit the ground
since h(t) represents the height of the rocket about the ground
when it hits the ground h(t) is equal to zero
therefore I want to solve the equation
zero is equal to a negative sixteen t squared
plus 80t plus 40
one way that i can solve this problem is to recognize that my three terms
have greatest common factor of negative eight therefore i'm going to factor out
a negative 8
this gives me a positive 2t squared
minus ten t
minus five
we could check that by using the distributive property
negative
eight times two t square it is a negative sixteen t squared
negative eight times a negative 10t is a positive 80t
and negative eight times a negative five is a positive forty
the next step that i am going to do is i'm going to
realize that i cannot factor my quadratic
equation
so i'm gonna use the quadratic formula
before i do that I'm going to divide both sides by a negative eight
zero divided by negative eight is still zero so i have zero is equal to 2t
squared
minus ten t
minus five
you may recall that
in the quadratic equation
ax squared plus bx plus c
my solution to the quadratic equation
can be given by the quadratic
quadratic formula
in this case since we were solving for t the quadratic formula would be given
by t
is equal to the opposite of b
plus or minus the square root of b squared minus 4ac
all over two times a
in this case t
is equal to
the opposite of a negative ten which is a positive ten
plus or minus the square root
of a negative ten squared
minus 4
times a
time c
all over two times a
now we just need to simplify the numerator and denominator
in the numerator we have 10
plus or minus the square root of negative ten squared
which is a hundred
and then we have
four times two which is eight
so a negative eight times a negative five is a positive forty
all over four
this is equivalent to ten
plus and minus the square root of a 140
all over four
taking this into my calculator and
noting that i want t to be equal to a positive number
I get t to be approximately 5.5 seconds
we can check to see if this makes sense by plugging in
5.5
knowing that we have some rounding error
when i do that plug in h(5.5)
and I get negative
four which means it really is below the ground
this gives me an approximation that 5.5 seconds is the correct answer