Question 1032193
i think i have the answer, but i needed to graph the equation it in order to find it because it is a very confusing problem.


i changed t to x and graphed y = 18.8 - 16.7 * cos((2pi/5)*x).


the graph is set to radians, since the angle is in radians.


the graph is shown below:


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when x is between 0 and 5, there are two points where the value of y is equal to 24.


those two points are when x = 1.502 and x = 3.498.


those values of x are the values of t that you are looking for.


you are looking for 2 points of intersection in the interval between x = 0 and x = 5.


this should be your solution, but it was arrived at graphically rather than algebraically.


i then determined how i would derive the same answers algebraically.


that proved a little more difficult, but i think i figured it out.


you start with the equation given.


that is h(t) = 18.8 - 16.7 * cos((2pi/5)*t). 


set h(t) = 24.


the equation becomes 24 = 18.8 - 16.7 * cos((2pi/5)*t).


subtract 18.8 from both sides of the equation to get 24 - 18.8 = - 16.7 * cos((2pi/5)*t).


simplify to get 5.2 = -16.7 * cos((2pi/5)*t).


divide both sides of this equation by -16.7 to get 5.2/-16.7 = cos(2pi/5)*t).


simplify to get -.3113772455 = cos((2pi/5)*t).


find the angle by taking the arc-cos (-.3113772455) to get the angle =  1.887438311.


that's the angle whose cosine is -.3113772455.


that angle is in the second quadrant because it is less than pi and greater than pi/2.


the cosine of that angle is negative, as it should be, because the cosine of an angle in the second quadrant is negative.


the cosine of an angle is negative in the second and third quadrant, so you are looking for an angle in the second and third quadrant.


you have the angle in the second quadrant.


you want to find the equivalent angle in the third quadrant.


the equivalent angle in the third quadrant is the angle in the third quadrant that will give you the same cosine.


the angle in the second quadrant is 1.887438311 radians.


the easiest way to find the equivalent angle in the third quadrant is to find the equivalent angle in the first quadrant and then convert that to the equivalent angle in the third quadrant.


the equivalent angle in the first quadrant for an angle that is in the second quadrant is equal to pi - 1.887438311 which is equal to 1.254154343 radians.


that's the equivalent angle in the first quadrant.


the equivalent angle in the third quadrant for an angle that is in the first quadrant is equal to pi + 1.254154343 radians.


that makes the equivalent angle in the third quadrant equal to 4.395746996 radians.


so you have two angles that will give you a cosine of -.3113772455.


those angle are 1.887438311 and 4.395746996 radians.


those angles are in the second and third qudrants, respectively.


each of those angles is equal to (2pi/5)*t


solve for t by dividing each of those angles by (2pi/5) and you will get:


t1 = 1.50197
t2 = 3.4980


round to the nearest hundredth to get:


t1 = 1.50
t2 = 3.50


your solutions are t = 1.50 and t = 3.50.


these solutions are confirmed graphically.


this was a difficult problem to visualize and i probably wouldn't have found the answer without graphing it first.


once i saw the graph and what the answer should be, i was able to logically deduce how to solve it graphically based on other aspects of trigonometry that i knew, like angles in different quadrants and when the cosine is negative, etc.