3) A flare is launched from a boat. The height, h, in meters, of the flare above the water is approximately modelled by the function h(t) = -15t2 + 150t, where t is the number of seconds after the flare is launched. How many seconds will it take for the flare to return to the water?
What is the highest point that the flare reaches?
To find out when the flare will return to the water, we need to set h(t) = 0 and solve for t.
-15t^2 + 150t = 0
Factor out a t:
t(-15t + 150) = 0
t = 0 or -15t + 150 = 0
t = 0 or t = 10
Since time cannot be negative, the flare will return to the water after 10 seconds.
To find the highest point that the flare reaches, we can find the vertex of the parabola -15t^2 + 150t. The vertex of a parabolic function of the form ax^2 + bx + c is given by (-b/2a, f(-b/2a)).
In this case:
a = -15
b = 150
The x-value of the vertex is -b/2a = -150 / 2*(-15) = 150 / 30 = 5
Plugging this back into the equation:
h(5) = -15(5)^2 + 150(5) = -15(25) + 750 = -375 + 750 = 375
Therefore, the highest point that the flare reaches is 375 meters above the water.