What is the value of z? 53=−19z+5+7z
2 answers:
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Problem 1)
Answer: pi/2
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Place your pencil at (0,3) which is a good starting point, since x = 0 is often a good starting point. Trace along the curve until you reach back up at y = 3 again. So you'll go down and then back up until you reach (pi/2, 3). The difference in x values is pi/2-0 = pi/2. Every pi/2 units, the graph curve repeats itself over and over infinitely.
Note: this is not the only way to determine the period. You can start at any point really. It helps to start at a min or max value, or at x = 0. In this case, it happened to be both.
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Problem 2)
Answer: See figure 2 (attached image)
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Point A is at (0,0)
Point B is at approximately (1.57,-2) which is exactly (pi/2,-2)
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Problem 3)
Answer: See figure 3 (attached image)
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Problem 4)
Answer: Choice B
The graph is horizontally compressed by a factor of 2 and shifts up 1 unit
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The jump from x to 2x will have the period go from 2pi to pi. The graph curve repeats itself more and more often (twice as often). Therefore we have an accordion squeeze effect going on, or think of it like a spring being pressed down. We have horizontal compression along the x axis.
Adding 1 to the end shifts every point upward 1 unit. Overall, this gives the visual the entire curve itself is shifted up 1 unit.
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Problem 5)
Answer: f(x) = 5.05*sin((pi/12)x)+5.15
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max = 10.2
min = 0.1
difference = 10.2-0.1 = 10.1
Divide this difference in half: 10.1 = 5.05
So a = 5.05 is the amplitude
The period is T = 24 hrs since the height is 5.15 at midnight of one day, it rises and then falls and rises back to 5.15 by the next midnight
b = 2pi/T = 2pi/24 = pi/12
c = 0 since there is no phase shift in this sine function
d = 5.15 is the midline, which is the starting height at x = 0
f(x) = a*sin(bx-c)+d
f(x) = 5.05*sin((pi/12)x-0)+5.15
f(x) = 5.05*sin((pi/12)x)+5.15
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Problem 6)
Answers are:
A) The max height of the Ferris wheel is 64 ft
B) The radius of the Ferris wheel is 30 ft
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The largest height of the table is 64, so 64 is the max height. This is why A is true.
B is true since the max is 64 and the min is 4, so 64-4 = 60 is the diameter, making the radius r = d/2 = 60/2 = 30
C is false because the height at t = 0 is 34 ft, but the lowest point is at height h = 4
D is false because the period is actually 10 seconds. Eg: going from t = 7.5 to t = 17.5 is a time of 10 seconds. During this time, the heights are 4, 34, 64, 34, 4 telling us that a full cycle has taken place.
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Answer: pi/2
-------------------------------
Place your pencil at (0,3) which is a good starting point, since x = 0 is often a good starting point. Trace along the curve until you reach back up at y = 3 again. So you'll go down and then back up until you reach (pi/2, 3). The difference in x values is pi/2-0 = pi/2. Every pi/2 units, the graph curve repeats itself over and over infinitely.
Note: this is not the only way to determine the period. You can start at any point really. It helps to start at a min or max value, or at x = 0. In this case, it happened to be both.
===========================================================
Problem 2)
Answer: See figure 2 (attached image)
-------------------------------
Point A is at (0,0)
Point B is at approximately (1.57,-2) which is exactly (pi/2,-2)
===========================================================
Problem 3)
Answer: See figure 3 (attached image)
===========================================================
Problem 4)
Answer: Choice B
The graph is horizontally compressed by a factor of 2 and shifts up 1 unit
-------------------------------
The jump from x to 2x will have the period go from 2pi to pi. The graph curve repeats itself more and more often (twice as often). Therefore we have an accordion squeeze effect going on, or think of it like a spring being pressed down. We have horizontal compression along the x axis.
Adding 1 to the end shifts every point upward 1 unit. Overall, this gives the visual the entire curve itself is shifted up 1 unit.
===========================================================
Problem 5)
Answer: f(x) = 5.05*sin((pi/12)x)+5.15
-------------------------------
max = 10.2
min = 0.1
difference = 10.2-0.1 = 10.1
Divide this difference in half: 10.1 = 5.05
So a = 5.05 is the amplitude
The period is T = 24 hrs since the height is 5.15 at midnight of one day, it rises and then falls and rises back to 5.15 by the next midnight
b = 2pi/T = 2pi/24 = pi/12
c = 0 since there is no phase shift in this sine function
d = 5.15 is the midline, which is the starting height at x = 0
f(x) = a*sin(bx-c)+d
f(x) = 5.05*sin((pi/12)x-0)+5.15
f(x) = 5.05*sin((pi/12)x)+5.15
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Problem 6)
Answers are:
A) The max height of the Ferris wheel is 64 ft
B) The radius of the Ferris wheel is 30 ft
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The largest height of the table is 64, so 64 is the max height. This is why A is true.
B is true since the max is 64 and the min is 4, so 64-4 = 60 is the diameter, making the radius r = d/2 = 60/2 = 30
C is false because the height at t = 0 is 34 ft, but the lowest point is at height h = 4
D is false because the period is actually 10 seconds. Eg: going from t = 7.5 to t = 17.5 is a time of 10 seconds. During this time, the heights are 4, 34, 64, 34, 4 telling us that a full cycle has taken place.
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There is a little-known theorem to solve this problem.
The theorem says that
In a triangle, the angle bisector cuts the opposite side into two segments in the ratio of the respective sides lengths.
See the attached triangles for cases 1 and 2. Let x be the length of the third side.
Case 1:
Segment 5cm is adjacent to the 7.6cm side, then
x/7.6=3/5 => x=7.6*3/5=4.56 cm
Case 2:
Segment 3cm is adjacent to the 7.6 cm side, then
x/7.6=5/3 => x=7.6*5/3=12.67 cm
The theorem can be proved by considering the sine rule on the adjacent triangles ADC and BDC with the common side CD and equal angles ACD and DCB.
The theorem says that
In a triangle, the angle bisector cuts the opposite side into two segments in the ratio of the respective sides lengths.
See the attached triangles for cases 1 and 2. Let x be the length of the third side.
Case 1:
Segment 5cm is adjacent to the 7.6cm side, then
x/7.6=3/5 => x=7.6*3/5=4.56 cm
Case 2:
Segment 3cm is adjacent to the 7.6 cm side, then
x/7.6=5/3 => x=7.6*5/3=12.67 cm
The theorem can be proved by considering the sine rule on the adjacent triangles ADC and BDC with the common side CD and equal angles ACD and DCB.