They are all different people, except just with the same faces. Emar is one of the boys in the building with a base of 1 meter. He is on the second floor at the very top of the stairway to the first floor. The horizontal distance between the floors is 20 m. The slope of the stairway is defined by the equation y=(-8/31)x + 8. Because Emar is nervous, he begins to walk fully up and down the stairway continuously. As he walks, his vertical height from his feet to the outside ground follow a sinusoidal pattern. On average, it takes him 75 seconds to get from one floor to the other. He begins to walk 2.5 minutes before the first selection is made.
Transcript
They are all different people, except just with the same faces.
Emar is one of the boys in the building with a base of 1 meter. He is on the second floor at the very top of the stairway to the first floor. The horizontal distance between the floors is 20 m. The slope of the stairway is defined by the equation y=(-8/31)x + 8. Because Emar is nervous, he begins to walk fully up and down the stairway continuously. As he walks, his vertical height from his feet to the outside ground follow a sinusoidal pattern. On average, it takes him 75 seconds to get from one floor to the other. He begins to walk 2.5 minutes before the first selection is made.
a) Determine the equation of his vertical height from his feet to the outside ground in terms of sine. Use seconds for your unit of time and meters for your unit of height. Assume that the point where Emar is standing has an x-coordinate of 0.5 on Cartesian Plane separate from the one involving the sinusoidal function.
*Neglect the actual time in your graph if you create one and make the starting time zero seconds. The outside ground has a y-value of zero as well.*
b) Assume Sophia, a girl, was already on the stairway before Emar began to walk. The horizontal distance between Emar and Sophia is 3.13 m. If she moves at a velocity of 14.3 m. every 75 seconds down the stairs, at what height will the two be at the same spot if they begin walking at the same time?
c) How many possible combinations of selected people are there if the time is 2:49 PM?
Part APart A
You should know that Part A is based on trigonometric You should know that Part A is based on trigonometric functions, so you use this basic equation.functions, so you use this basic equation.
Parameter A determines the amplitude and can be both Parameter A determines the amplitude and can be both negative and positive. The amplitude is always positive negative and positive. The amplitude is always positive though. It deals with the vertical stretch. Parameter B though. It deals with the vertical stretch. Parameter B deals with the horizontal stretch. Parameter C deals deals with the horizontal stretch. Parameter C deals with the horizontal translation. Parameter D deals with with the horizontal translation. Parameter D deals with the vertical translation.the vertical translation.
Conceptualize TimeConceptualize Time
Emar is at the top of the stairway to the first floor. The slope of the stairway is defined by the equation shown. The horizontal distance is the distance that deals only with the x components. In this case, it is the distance between the x-coordinates of his position and end of the stairway (start of floor 1)
What we must do is find the vertical distance between the floors. The vertical distance is indicated by the yellow line. It deals with the y components of Emar’s position and end of the stairway. We need the vertical distance because it deals with Emar’s height to the ground as does the base.
Finding the Vertical Distance IFinding the Vertical Distance I
We use the equation of the stairway’s slope to determine the y-coordinates of Emar’s position and end of stairway. We plug in the position’s respective x-values to determine their y-values. We know that the horizontal distance between the floors is 20 m, so we add 20 to ½. We do not use 20 as the x-value because the stairway is only a portion of the graph that begins when x = 1/2.
Finding the Vertical Distance IIFinding the Vertical Distance II
Now we subtract the y-values to obtain the distance between the floors. All you do is subtract one from the other and obtain the absolute value of the resultant value. This is the vertical distance just between the floors and is not the final vertical distance since this does not account for the base of the building. The base is 1 meter thick and the question wants you to determine Emar’s height from his feet to the outside ground. All you do is add 1 to the vertical distance you found. You should know that the y-value of the outside ground is zero. This was stated earlier in the question.
Change in Y = (244/31) – (84/31)Change in Y = 160/31Therefore, the vertical distance between the floors is 160/31 meters
Vertical Distance From Max. Height. To Min. Height(160/31) + 1 = 191/31
We add 1 because the base is 1 meter thick
We want the height indicated by the pink line
This is an alternate way to find the Vertical DistanceThis is an alternate way to find the Vertical Distance
In the graph, there is a red pattern to show the stuff we ignore. We are focusing only on the triangle since we are going to use the Pythagorean Theorem. First, we find the distance between the points, which is amazingly the hypotenuse. We should have two values, the hypotenuse and horizontal component, 20 not 20.5. Remember, we are focusing only on the triangle. The horizontal distance between the known vertices is 20.
P ( 41/2 , 84/31 )P ( ½ , 244/31 )
D = squareroot[ ( ½ -41/2 )^2 + ( 244/31 – 41/2 )^2 ]
Final Distance = Thickness of Base + Vert. Dis. Between Floors = 1 + 5.1612903 = 6.1612903This decimal number is approximately equal to 191/31, the fraction we obtained earlier.
Parameter AParameter A
This value, 191/31, is the vertical distance between the top of the stairway and ground level. You should know that 191/31 is also the maximum height he achieves. The minimum height he achieves is 1 because the base of the building is 1 meter thick. His feet don’t go through the base obviously.
To find parameter A, we subtract the maximum and minimum height from each other and divide that answer by two.
This value is parameter A. Actually, parameter is also -80/31. I’ll show you this later.
Current EquationCurrent Equation
So far, our equations are:
H(t) represent the height as a function of time in metersT represents time in seconds
Parameter BParameter B
The information states that his height follows a sinusoidal wave. It is obviously not a tangential wave. We also know that it takes 75 seconds for him to get from one floor to the other. One complete cycle is from one floor to the other, then back to the previous floor. It must take 75 seconds x 2 to complete one cycle then. This is what we do
Period for one cycle = 75 seconds x 2Period for one cycle = 150 seconds
Parameter B = 2Π / periodParameter B = 2Π / 150
The reason why we use 2pi is because the standard cosine graph has a period of 2Π. A tangent graph has a period of just Π. Parameter B does not need to be simplified. Leaving it this way actually helps more.
Current EquationCurrent Equation
So far, our equations are:
H(t) represent the height as a function of time in metersT represents time in seconds
Finding Parameter C with a positive Parameter AFinding Parameter C with a positive Parameter A
Parameter C for this equation is different from the equation containing a negative parameter A.
These dots show his height in one cycle at equal time intervals. It makes a cosine shape.
This is what a sine shape looks like compared to a cosine shape. To make this shape equal to the shape at the top, we shift it to the left.
You can see that we just shift it to the left by 37.5 seconds. This gets rid of the -37.5 seconds. To make this more clear, if shifted the shape on the right inside a picture frame, the dot at -37.5 seconds would eventually get cropped off.
Parameter C = - 37.5
Finding Parameter C with a negative Parameter AFinding Parameter C with a negative Parameter A
Like earlier, the graph at the left is what we must achieve. In order to do that, we can shift the graph on the right to the left by 112.5 seconds. Or we can shift the graph to the right by 37.5 seconds.
If it goes into the left direction, the value is negative. However, in the equation watch out because the formula uses a negative symbol.
Parameter C = + 37.5 or - 112.5 seconds
Current EquationCurrent Equation
So far, our equations are:
H(t) represent the height as a function of time in metersT represents time in seconds
Parameter DParameter D
Parameter D is the y-value halfway between the maximum and minimum values. It is also known as the sinusoidal axis.
What you do is subtract the amplitude (always positive) from the maximum height. Or you can add the amplitude to the minimum height. Either way, the resultant value is the same
(191/31) – (80/31) = 111/31
Or
1 + (80/31) = 111/31
Current and Final EquationCurrent and Final Equation
So far, our equations are:
H(t) represent the height as a function of time in metersT represents time in seconds
Part BPart B
The question earlier stated that it takes Sophia 75 seconds to travel 14.3 meters. Therefore, we can deduce that her velocity is:
14.3 meters / 75 seconds
From earlier, we determined the length of the hypotenuse of the line. It takes Emar 75 seconds to go down the stairs and 75 seconds to go up the stairs., The length of the stairs is 20.65523948 meters. Therefore, we can deduce that his velocity is:
20.65523948 meters / 75 seconds
Finding the Distance Between the TwoFinding the Distance Between the Two
You might have guessed that this question involves the equation:Distance = velocity x time
Well, that is what we are going to use. Let’s find the distance between the two people. We will first obtain their coordinates using the equation from earlier.
Hypotenuse = Their Distance From Each OtherHypotenuse = Their Distance From Each Other
We use the distance formula to obtain the distance between the two people. Now this is what we know:
Distance Between Two = 3.232544977 metersSophia’s Velocity = 14.3 meters / 75 secondsEmar’s Velocity = 20.65523948 meters / 75 seconds
D = squareroot[ ( ½ -3.63 )^2 + ( 244/31 – 218.96/31 )^2 ]
D = 3.232544977 = hypotenuse
P( ½ , 244/31) P( 3.63 , 218.96/31)
Understanding Distance, Time and VelocityUnderstanding Distance, Time and Velocity
E represents Emar’s position on the stairs.S represents Sophia’s position on the stairs.T is an arbitrary spot chosen for where they will meet.
Distance Between Two = 3.232544977 metersSophia’s Velocity = 14.3 meters / 75 secondsEmar’s Velocity = 20.65523948 meters / 75 seconds
ES
T3.232544977 meters
What we do is solve for the distances between the letters. We already have ES. Now we just needET and ST.You should know that distance = velocity x time. Therefore, we can say this:ES = 3.232544977 metersET = (20.65523948 meters / 75 seconds) x TimeST = (14.3 meters / 75 seconds) x Time
Understand that ET = ES + ST
Finding TimeFinding Time
We can plug our information into this equation then. Let T = Time
ET = ES + ST(20.65523948 meters / 75 seconds) x T = 3.232544977 meters + (14.3 meters / 75 seconds) x T
Isolate 3.232544977 meters
3.232544977 meters = [ (20.65523948 meters / 75 seconds) x T ] – [(14.3 meters / 75 seconds) x T ]
ES = 3.232544977 metersET = (20.65523948 meters / 75 seconds) x TimeST = (14.3 meters / 75 seconds) x Time
ET = ES + ST
Finding the heightFinding the height
Now we plug in T into any of the three equations we obtained earlier. I will use two of them to prove that the equations are correct. The heights should be the same if everything is correct.
T = 38.14818844 seconds
Here is an alternate way to find the height, my way.Here is an alternate way to find the height, my way.
Find the velocity with the greatest value. We subtract Sophia’s velocity from Emar’s velocity.
We subtract the two because they are traveling in the same direction. Relative velocity is the velocity of one object relative to another. If we were to add the two, then the two would be traveling in the same direction. To further clear this up, I’m traveling at 10m/s while you’re traveling at 5m/s. My relative velocity to you is 5m/s because it is as if I’m traveling at 5m/s and you’re at rest. In reality, you’re not but that is basically what relative velocity is. If we were heading towards each other, both of our relative velocities would be 15m/s. We’re not actually reaching those speeds, but the time it takes to reach each other is the same if I was at rest and you were moving 15m/s or if we were moving 5m/s towards me and I was moving 10m/s towards you.
Here is an alternate way to find the height, my way.Here is an alternate way to find the height, my way.
Find the velocity with the greatest value. We subtract Sophia’s velocity from Emar’s velocity.
From my explanation earlier, you should know that it is as if Sophia is at rest while I’m moving at a velocity of 6.35523948 meters / 75 seconds.
So because we are saying she is at rest and her distance from me is 3.232544977 meters, we can solve for Time using distance = velocity x time.
3.232544977 meters = (6.35523948 meters / 75 seconds) x Time
Our answer is the same as earlier.
Time = 38.14818844 seconds
Finding the heightFinding the height
Now we plug in T into any of the three equations we obtained earlier. I will use two of them to prove that the equations are correct. The heights should be the same if everything is correct.
T = 38.14818844 seconds
Part CPart C
First of all, we must determine the number of people that should have already been selected when the time is 2:49PM. Here is a table to determine this.
Remember that selections begin 2.5 minutes after 2:35 PM. In addition, there is a 2.5 minute interval between each selection.
2:49 is between 2:47.5 and 2:50. Therefore, 5 people are already chosen. We can’t choose a number in between 5 and 6 because we can’t split people. They only count as wholes.
We also know that either a boy or girl could be chosen first. This is something to put into consideration.
Part CPart C
Part CPart C
The denominator is the factorial of number of slots for each case.
Part CPart C
Congratulations! You finished Congratulations! You finished Question 2!Question 2!
