The height h, in metres, above the ground of a car as a ferris wheel rotates can be modelled by the function h(t) = 18cos(πt/80) +19
what is the minimum height of a car?
do i like subtract 19 from 18 ?
what is the maximum height of a car?
do i add the 19 to 18?
how long does it take for the wheel to revolve once?
80s?
The question below is a different question..
The period in degrees of the graph y = cos4x is?
Is it 45 degrees? I don't get how to solve this?
Just look at the trig part of the function, namely the
cos(....)
the min value of that is -1 and the maximum is +1
so 18cos(..) has a min of -18 and a max of +19
so -18 + 19 = 1 metre, ---> minimum height of car
and +18 + 19 = 37 ----> maximum height of car
for y = a cos kx
the period is 2π/k or 360°/k
your value of k is π/80 , you had πt/80 which is (π/80)t
so the period is 2π /(π/80)
= 2π(80/π) = 160
So the period is 160 seconds , if t is defined in seconds
for the 2nd question:
y = cos 4x ----> compare with y = cos kx
period = 2π/4 radians or 360°/4
= π/2 radians or 90°
Memorize: period = 2π/k or 360°/k
Ah, the joys of ferris wheels and trigonometry! Let's tackle these questions one by one:
Firstly, to find the minimum height of the car on the ferris wheel, you don't subtract 19 from 18. Instead, you need to remember that the minimum value of the cosine function is -1, so the minimum height is actually 18*(-1) + 19. Can you guess what that is? Yep, it's 1 meter above the ground!
Now, for the maximum height, you've got it a little mixed up. To find the maximum height of the car, you need to add 19 to the amplitude of the cosine function, which is 18. So the maximum height is 18 + 19 = 37 meters. Just imagine the view from up there!
As for how long it takes for the ferris wheel to revolve once, you're on the right track! The period of the cosine function, which represents one full revolution or cycle, is found by taking the denominator of the t inside the cosine function. So in this case, the period is indeed 80 seconds. That's a whole lot of spinning!
Now, for your other question about the period in degrees of the function y = cos(4x), it's a bit different. In this case, we're dealing with degrees, not seconds. The period of a cosine function in degrees is given by dividing 360 degrees by the coefficient of x, which is 4 in this case. So the period here is 360/4 = 90 degrees.
I hope that clears things up for you! If you need more amusement, just give me a holler!
To find the minimum height of the car, you need to subtract 19 from the expression 18cos(πt/80). This represents the lowest point the car reaches during one rotation of the ferris wheel.
So, the minimum height of the car is 18 - 19 = -1 meter (below the ground level).
To find the maximum height of the car, you need to add 19 to the expression 18cos(πt/80). This represents the highest point the car reaches during one rotation of the ferris wheel.
So, the maximum height of the car is 18 + 19 = 37 meters.
Now, let's find out how long it takes for the wheel to revolve once. The function h(t) = 18cos(πt/80) + 19 is periodic with a period of 2π/π = 2.
Therefore, it takes 2 units of time (let's assume seconds) for the wheel to complete one revolution.
Now, for the second question:
To find the period in degrees of the graph y = cos4x, you need to determine the length of one full cycle of the graph.
The period of a function is given by 2π divided by the coefficient of x. In this case, the coefficient of x is 4.
So, the period in radians (not degrees) is 2π/4 = π/2.
To convert this to degrees, we multiply by 180/π.
Therefore, the period in degrees is (π/2) * (180/π) = 90 degrees.
So, the period in degrees of the graph y = cos4x is 90 degrees.
To find the minimum and maximum height of the car, you need to consider the amplitude of the cosine function. In the given function h(t) = 18cos(πt/80) + 19, the constant 18 represents the amplitude.
The minimum height of the car would be when the cosine function reaches its lowest value, which is -1. Therefore, you need to subtract the amplitude (18) from the constant term (19) to find the minimum height.
Minimum height = 19 - 18 = 1 meter
The maximum height of the car would be when the cosine function reaches its highest value, which is +1. Therefore, you need to add the amplitude (18) to the constant term (19) to find the maximum height.
Maximum height = 19 + 18 = 37 meters
Now, to determine the time it takes for the Ferris wheel to revolve once, you can look at the period of the cosine function. The period is the time it takes for the function to complete one full cycle. In the given function, the period is equal to the length of one complete cycle, which can be calculated as:
Period = 2π / (π/80) = 2 * 80 = 160 seconds
So, it takes 160 seconds for the wheel to revolve once.
Moving onto the next question:
To find the period in degrees of the function y = cos(4x), you need to understand that the period of a cosine function is determined by the coefficient attached to the x value in the argument of cos.
In this case, the coefficient is 4. The general formula to calculate the period in degrees is:
Period (in degrees) = 360 / Absolute value of the coefficient
Applying this formula to the given function:
Period (in degrees) = 360 / 4 = 90 degrees
Therefore, the period in degrees of the graph y = cos(4x) is 90 degrees.