Answer:
Part A) The potato hit the ground at t=5.70 seconds (see the explanation)
Part B) The potato is 40 feet off the ground at the time t=5.28 seconds (see the explanation)
Step-by-step explanation:
we have
[tex]h(t)=-16t^2+80t+64[/tex]
where
h(t) is the height of a potato in feet
t is the time in seconds
Part A) Write an equation that can be solved to find when the potato hits the ground. Then solve the equation
we know that
When the potato hit the ground, the value of h(t) must be equal to zero
so
For h(t)=0
[tex]-16t^2+80t+64=0[/tex]
Solve the quadratic equation
The formula to solve a quadratic equation of the form
[tex]ax^{2} +bx+c=0[/tex]
is equal to
[tex]x=\frac{-b\pm\sqrt{b^{2}-4ac}} {2a}[/tex]
in this problem we have
[tex]-16t^2+80t+64=0[/tex]
so
[tex]a=-16\\b=80\\c=64[/tex]
substitute in the formula
[tex]t=\frac{-80\pm\sqrt{80^{2}-4(-16)(64)}} {2(-16)}[/tex]
[tex]t=\frac{-80\pm\sqrt{10,496}} {-32}[/tex]
[tex]t=\frac{-80+\sqrt{10,496}} {-32}=-0.70[/tex]
[tex]t=\frac{-80-\sqrt{10,496}} {-32}=5.70[/tex]
therefore
The potato hit the ground at t=5.70 seconds
Part B) Write an equation that can be solved to find when the potato is 40 feet off the ground. Then solve the equation
For h(t)=40 ft
substitute in the quadratic equation
[tex]-16t^2+80t+64=40[/tex]
[tex]-16t^2+80t+24=0[/tex]
Solve the quadratic equation
we have
[tex]a=-16\\b=80\\c=24[/tex]
substitute in the formula
[tex]t=\frac{-80\pm\sqrt{80^{2}-4(-16)(24)}} {2(-16)}[/tex]
[tex]t=\frac{-80\pm\sqrt{7,936}} {-32}[/tex]
[tex]t=\frac{-80+\sqrt{7,936}} {-32}=-0.28[/tex]
[tex]t=\frac{-80-\sqrt{7,936}} {-32}=5.28[/tex]
therefore
The potato is 40 feet off the ground at the time t=5.28 seconds
Given the equation that models the height of the potato given as h(t) = -16t² + 80t + 64, the potato hits the ground at when h(t) = 0
The equation becomes -16t² + 80t + 64 = 0
Factorizing there result:
16t² - 80t - 64 = 0
t² - 5t - 4 = 0
On factorizing the value of t
t = 5±√25-4(-4)/2
t =5±√25+16/2
t=5±√41/2
t = 5+6.4/2 and 5 - 6.4/2
t = 5.7secs
Hence the potato will hit the ground after 5.6 secs
b) If the potato is 40 feet off the ground, the equation becomes
-16t² + 80t + 64 = 40
Factorizing there result:
16t² - 80t - 24 = 0
2t² - 10t - 3 = 0
On factorizing the value of t
t = 10±√100-4(-3)/2
t =10±√100+12/2
t=10±√112/2
t = 10+10.6/2 and 10 - 10.6/2
t = 10.3 secs
The potato will be 40 feet above the ground after 10.3 secs
Learn more on factorization here: https://brainly.com/question/25829061
