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A ship fires a distress flare up into the air. The height (meters) and the time (seconds)
are related by y = - 5 ( x - 2.5) ² + 75
(a) What was the maximum height of the flare?
(b) At what time (in seconds) did the maximum height occur?
(c) At what height was the flare fired from the ship?
(d) When did the flare hit the water?
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The formula y = -5*(x-2.5)^2 + 75 represents a downward parabola,
written in vertex form.
(a) The maximum height is 75 meters
(because some non-negative quantity 5*(x-2.5)^2 is subtracted from 75).
(b) The maximum height of 75 meters occurs when x= 2.5 seconds, making the term -5*(x-2.5) equal to zero.
(c) To answer this question, substitute x= 0 into the formula, which
represents the initial condition when the time counter started.
You will get y = -5*(0-2.5)^2 + 75 = -5*2.5^2 + 75 = -5*6.25 + 75 = 43.75 meters.
(d) To answer this question, equate the height to zero
-5*(x-2.5)^2 + 75 = 0.
Simplify step by step, solving this quadratic equation, and find x
-5*(x-5)^2 = -75;
(x-5)^2 = (-75)/(-5) = = 15;
x - 5 = sqrt(15) = 3.873 seconds;
x = 3.873 + 5 = 8.873 seconds.
Solved in full.
