At the circus, a poodle is shot out of a cannon. The height above the ground (in feet) of the dog after t seconds is given by `f(t)=-16t^2+78t+6` . Round answers to the nearest hundredth. a. When...

At the circus, a poodle is shot out of a cannon. The height above the ground (in feet) of the dog after t seconds is given by `f(t)=-16t^2+78t+6` . Round answers to the nearest hundredth.


a. When will the poodle land in a pool of water at ground level?


b. Give the time interval that the poodle’s height is more than 20 ft.


c. What is the poodle’s maximum height during flight?

Expert Answers
tjbrewer eNotes educator| Certified Educator

a) `-16t^2+78t+6=0` we find t value by applying the quadratic formula `t=(-78+-sqrt(78^2-4(-16)(6)))/((2)(-16))=(-78+-sqrt(6084+384))/(-32)=`

`(-78+-sqrt(6468))/(-32)=(-78+-80.42)/(-32)=2.42/-32, -158.42/-32=-.08, 4.95` Common sense tells us that t cannot have a negative value, so t=4.95 sec. 

b)` ``-16t^2+78t+6>20` We solve this by subtracting 20 from both sides, and then re-applying the quadratic formula to the inequality `-16t^2+78t-14>0` The interval is between the two values of t for `t=(-78+-sqrt(78^2-4(-16)(-14)))/((2)(-16))=(-78+-sqrt(6084-896))/(-32)=`

`(-78+-sqrt(5188))/-32=(-78+-72.03)(-32)=(-5.97)/-32, (-150.03)/-32=`

`0.19, 4.69`So the poodle's height exceeds 20 ft from 0.19sec, to 4.69 sec. 

c) The formula for the Max/min of a quadratic equation is `t=-(78/(-32))=2.44` The poodle will reach max height at 2.44sec, and plugging that value of t into our equation finds a maximum height.  `f(2.44)=-16(2.44)^2+78(2.44)+6=`

`-95.26+190.32+6=` 101.06 ft.