SOLUTION: Sean and Mason run out of gas while fishing in the bay. They set off an emergency flare with an initial vertical velocity of 30 meters per second. The height of the flare in mete

Algebra ->  Customizable Word Problem Solvers  -> Mixtures -> SOLUTION: Sean and Mason run out of gas while fishing in the bay. They set off an emergency flare with an initial vertical velocity of 30 meters per second. The height of the flare in mete      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 1170724: Sean and Mason run out of gas while fishing in the bay. They set off an emergency flare with an initial vertical velocity of 30 meters per second. The height of the flare in meters can be modeled by h(t)=-5t^2+30t, where t represents the number of seconds after launch. They think the flare has to reach a height of 15 meters to be seen from the shore.
a.How long will the flare be 15 m high or higher?
b.When will the flare hit the water?
c.How high will the flare be after 2 seconds?
d.How high will the flare be after 5 seconds?
e.What is the maximum height the flare will reach? f.How long will it take for that max to occur?
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
a. How long will the flare be 15m high or higher?
h%28t%29%3E=15
+-5t%5E2%2B30t+%3E=15.....simplify
+-t%5E2%2B6t+%3E=3
+-t%5E2%2B6t-3+%3E=0
+t%3E=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a
+t%3E=%28-6%2B-sqrt%286%5E2-4%28-1%29%28-3%29%29%29%2F-2
+t%3E=%28-6%2B-sqrt%2824%29%29%2F-2
+t%3E=%28-6%2B-4.9%29%2F-2
need only positive solution
+t%3E=%28-6-4.9%29%2F-2
+t%3E=5.5seconds


b.When will the flare hit the water?
h%28t%29=0
+-5t%5E2%2B30t=0
+-t%5E2%2B6t=0
+-t%28t-6%29=0
need only positive solution
t=6 seconds

c. How high will the flare be after 2 seconds?
t=2
h%282%29=-5%2A2%5E2%2B30%2A2
+h%282%29=-20%2B60
+h%282%29=40


d.How high will the flare be after 5+seconds?
t=5
+h%285%29=-5%2A5%5E2%2B30%2A5
+h%285%29=-125%2B150
+h%285%29=25
e.What is the maximum height the flare will reach?
the maximum height will be at vertex, write equation in vertex form
h%28t%29=-5t%5E2%2B30t......complete square
h%28t%29=-5%28t%5E2-6t%29
h%28t%29=-5%28t%5E2-6t%2Bb%5E2%29-%28-5b%5E2%29
h%28t%29=-5%28t%5E2-6t%2B3%5E2%29%2B5%2A3%5E2
h%28t%29=-5%28t-3%29%5E2%2B45
=> vertex is at (3,45)
the maximum height will be 45meters

f. How long will it take for that max to occur?
the maximum height the flare will reach in 3 seconds


Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

            In the post by  @MathLover1,  the solution to part  (a)  is incorrect.

            See my correct solution below. 


a. How long will the flare be 15 m high or higher?


   For it, this inequality must be held

       -5t^2 + 30t >= 15


    Simplify

       -5t^2 + 30t - 15 >= 0


    Change the signs by multiplying both sides by (-1)

        5t^2 - 30t + 15 <= 0    


    Cancel the factor 5

         t^2 - 6t + 3 <= 0


    +-----------------------------------------------------------------+
    |    Physically, it is OBVIOUS that the time should be between    |
    |    two time moments when the height is exactly 15 ft.           |
    +-----------------------------------------------------------------+


    According to it, we should determine these time moments from the equation

        t^2 - 6t + 3 = 0    


    Solve using the quadratic formula 

        t%5B1%2C2%5D = %286+%2B-+sqrt%286%5E2+-+4%2A3%29%29%2F2 = %286+%2B-+sqrt%2824%29%29%2F2 = 3+%2B-+sqrt%286%29 = 3+%2B-+2.45.

        t%5B1%5D = 3 - 2.45 = 0.55;  t%5B2%5D = 3 + 2.45 = 5.45.


    So, the ANSWER is  0.55 <= t <= 5.45  seconds.


    For better understanding, see the plot below 



    


    Plot  y = -5t^2 + 30t (red)  and  y = 15 (green)