If we just have
![y = \cos x](https://tex.z-dn.net/?f=y%20%3D%20%5Ccos%20x)
that's an amplitude of 1 and a period of ![2\pi](https://tex.z-dn.net/?f=2%5Cpi)
Let's modify this step by step.
Amplitude of 4. Four times the amplitude means a factor of 4 on the outside.
![y = 4 \cos x](https://tex.z-dn.net/?f=y%20%3D%204%20%5Ccos%20x)
Period of π. Double the period means a factor of 2 on the inside.
![y = 4 \cos(2x)](https://tex.z-dn.net/?f=y%20%3D%204%20%5Ccos%282x%29)
Horizontal shift of π/2 to the left. To the left means adding a positive number to x. The question appear ambiguous. It's not clear to me if we add to x or 2x; let's make it
![y = 4 \cos(2(x + \frac \pi 2))](https://tex.z-dn.net/?f=y%20%3D%204%20%5Ccos%282%28x%20%2B%20%5Cfrac%20%5Cpi%202%29%29)
Vertical shift of 3:
![y = 3 + 4 \cos(2(x + \frac \pi 2))](https://tex.z-dn.net/?f=y%20%3D%203%20%2B%204%20%5Ccos%282%28x%20%2B%20%5Cfrac%20%5Cpi%202%29%29)
![y = 3 + 4 \cos(2x + \pi)](https://tex.z-dn.net/?f=y%20%3D%203%20%2B%204%20%5Ccos%282x%20%2B%20%20%5Cpi%29)
Depending on the interpretation of the phase shift, that pi may be a pi/2.
Answer: ![3 + 4 \cos(2x + \pi)](https://tex.z-dn.net/?f=3%20%2B%204%20%5Ccos%282x%20%2B%20%20%5Cpi%29)
Answer:
c × 0.90
Step-by-step explanation:
c × 0.90, would be your answer because 10% is equal to 0.10, then you would subtract the 0.10 (10%) by 1 (100%), then multiply c times the 0.90 that you get, to get the decreased number, because you're multiplying by a decimal
Answer: c) 2x-5
We can represent two times micheal’s age as 2x, and 5 years younger would be you subtract 5
So the answer is 2x-5
There would be a 1/36 chance of rolling a 6 then a 3 consecutively on a 6-sided die
Because the chance of rolling each of the numbers individually would be 1/6, and you are trying to roll 2 specific numbers in a row on a single die, you can simply multiply 1/6 by 1/6 to get your answer.
Here, it is helpful to use the formulae for the volume of a cone and a cylinder. I am assuming we are dealing with a right cone and a right cylinder.
![V_{cone}= \frac{1}{3} \pi r^2h](https://tex.z-dn.net/?f=V_%7Bcone%7D%3D%20%5Cfrac%7B1%7D%7B3%7D%20%20%5Cpi%20r%5E2h)
and
![V_{cyl}= \pi r^2h](https://tex.z-dn.net/?f=V_%7Bcyl%7D%3D%20%5Cpi%20r%5E2h)
.
The volumes of both of these figures are equal to 34 cubic inches, as you said. Notice in our formulae, everything is identical EXCEPT that the volume of a cone is basically that of a cylinder divided by three. To reverse this, that is, to find the volume of the cylinder, we would multiply by 3. 34 cubic inches times 3 is
102 cubic inches.
I would like to add that sometimes these formulae seem totally arbitrary. But when you think about the cylinder like a circle with a rolled-up rectangle, you can see that the
![\pi r^2](https://tex.z-dn.net/?f=%20%5Cpi%20r%5E2)
part is the area of a circle, and the height is because it's three dimensional. To translate this into the volume of a cone is a bit trickier. It involves calculus...
![\int\ {x^2} \, dx = \frac{1}{3}x^3](https://tex.z-dn.net/?f=%20%5Cint%5C%20%7Bx%5E2%7D%20%5C%2C%20dx%20%3D%20%20%5Cfrac%7B1%7D%7B3%7Dx%5E3)
. If that looks like nonsense to you, then you can just remember that a cone is kind of a pointy cylinder and must be smaller than a cylinder!