Find the reference angle for a rotation angle of 5pi/6 radians.
1 answer:
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Answer:
19 million
Step-by-step explanation:
So generally whenever we express a exponential function, it's in the form:
where a=initial value aka when x=0
in this case, b can be expressed as: , where r is the growth rate, which in this case is 2.13%, and expressing that as a decimal (divide by 100), r=0.0213, thus b=1.0213
This gives us the following equation:
To kind of give you a intuitive reason as to why this makes sense, think of that 1.0213 wrapping around the previous value.
So when x=0, it's actually just 16, since any number (besides 0) to the power of 0, equals 1.
When x=1, we get: , which can be expressed as a 2.13% increase from year 0
When x=2, we get: , which can be expressed as a 2.13% increase from year 1
when x=3, we get: , which can be expressed as a 2.13% increase from year 2.
In other words, each year we're getting another 1.0213 to multiply by (very definition of an exponent).
So to calculate the population in 1997, we put in x=7, since it's 7 years after 1990
We get the following equation:
Using a calculator this simplifies to approximately:
Rounding this to the nearest integer would yield 19, and since the values are expressed in millions, the answer would be 19 million
Answer:
Summation notation is:
If you prefer it a little more simplified:
Step-by-step explanation:
First my favorite part, finding a pattern between the consecutive terms.
This is an arithmetic series because the terms are going up by 5 each time.
So arithmetic sequence, think linear equations:
x | y
1 -9
2 -4
3 1
4 6
..................
n 66
We are going to have to find that n but will will eventually...
The equation for a line in point slope form is where
is a point on the line and m is the slope.
We are already have the slope is 5 (the slope is the common difference in arithmetic sequence).
I'm going to use the first point (1,-9).
So the equation in point slope form is
Subtract 9 on both sides:
Now we need to know how many terms we are adding so what is x if y=66.
Add 9 on both sides:
Divide both sides by 5:
Add 1 on both sides:
We have 16 terms in this sequence where the 16th term is 66.
Summation notation is:
You could simplify the 5(x-1)-9.
Distribute: 5x-5-9
Add like terms: 5x-14