Question:
a sea bird dives off a cliff into the sea. it's vertical height (h) above sea lievel in meters is given by the equation h = t^3 - 6t^2 - t +30, where t is the time in seconds since the dive.
1) show that this cubic equation can be written as h = (t+2 (t-3) (t-5)
b) at what times is the bird at sea level?
c) how long is the bird under the water?
how do i go about solving these equations? i'm finding this question really hard
thank you so much!
a sea bird dives off a cliff into the sea. it's vertical height (h) above sea lievel in meters is given by the equation h = t^3 - 6t^2 - t +30, where t is the time in seconds since the dive.
1) show that this cubic equation can be written as h = (t+2 (t-3) (t-5)
b) at what times is the bird at sea level?
c) how long is the bird under the water?
how do i go about solving these equations? i'm finding this question really hard
thank you so much!
-
a)
(t + 2) (t - 3) (t - 5) = (t - 2) (t^2 - 8t + 15)
. . . . . . . . . . . . . . = t (t^2 - 8t + 15) + 2 (t^2 - 8t + 15)
. . . . . . . . . . . . . . = t^3 - 8t^2 + 15t + 2t^2 - 16t + 30
. . . . . . . . . . . . . . = t^3 - 6t^2 + - t + 30
b)
h(t) = 0
(t + 2) (t - 3) (t - 5) = 0
t = 3 or 5 seconds
(Since t is time since dive, we cannot have t = -2, since t must be >= 0)
c)
For 0 < t < 3, h > 0
For 3 < t < 5, h < 0
For t > 5, h > 0
So h < 0 on interval 3 < t < 5
So bird is under the water for 2 seconds
(t + 2) (t - 3) (t - 5) = (t - 2) (t^2 - 8t + 15)
. . . . . . . . . . . . . . = t (t^2 - 8t + 15) + 2 (t^2 - 8t + 15)
. . . . . . . . . . . . . . = t^3 - 8t^2 + 15t + 2t^2 - 16t + 30
. . . . . . . . . . . . . . = t^3 - 6t^2 + - t + 30
b)
h(t) = 0
(t + 2) (t - 3) (t - 5) = 0
t = 3 or 5 seconds
(Since t is time since dive, we cannot have t = -2, since t must be >= 0)
c)
For 0 < t < 3, h > 0
For 3 < t < 5, h < 0
For t > 5, h > 0
So h < 0 on interval 3 < t < 5
So bird is under the water for 2 seconds