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The height (in meters) of a shot cannonball follows a trajectory given by h(t)=-4.9t^2+14t-0.4  at time t  (in seconds). For how many seconds is the height of the cannonball at least 6 meters?

Determine the set of all real x satisfying (x^2+3x-1)^2<9

Expressing your answer in interval notation, find all values of x such that x(x-2)^2 * (x+1)<0.

THANK YOU

Nov 3, 2019

#1
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(1) Solving the inequality, the answer is 5 seconds.

(2) Solving the inequality, the answer is (-5,2).

Nov 3, 2019
#2
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h(t)=-4.9t^2+14t-0.4       substitute  6 in for h(t) and solve for t

you should get two VALUES for t   the ball is above 6m for the time in between

6 = -4.9t^2+14t-.4

-4.9t^2 +14t- 6.4 =0    solve for t = 0.571429   and   2.28571    now you calc the time the ball is above 6m

Nov 3, 2019
#3
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Determine the set of all real x satisfying (x^2+3x-1)^2<9

Take both roots and we have that

x^2 + 3x - 1   =    3                     x^2   + 3x - 1  = - 3

x^2  + 3x  - 4   = 0                     x^2  +  3x  + 2   =   0

(x + 4) (x - 1)  = 0                     (x  + 2)  ( x + 1)   =  0

Setting each factor to 0 and solving for x  gives the answers that  x  = { -4, - 2,- 1, 1 }

(-inf, -4)    ( -4, -2)   ( -2, -1)   (-1, 1)    or  ( 1 inf )

Testing a point in each interval in the original inequality we see that the  solution intervals are

(-4, -2)  U   (-1, 1)   Nov 3, 2019