The number of dairy cows in a farm can vary from month to month. The amount of milk produced each month depends on the number of dairy cows on the farm. If g is the number of gallons of milk produced and n is the number of dairy cows on the farm, which of the following statement or statements are true about the variables?
The statement "The amount of milk produced each month depends on the number of dairy cows on the farm" is true. This means that the amount of milk produced, represented by g, is dependent on the number of cows, represented by n.
Therefore, we can say that:
- g is a dependent variable, as it depends on the value of n.
- n is an independent variable, as its value is not dependent on g or any other variable.
- There is a functional relationship between g and n, as the value of g can be expressed as a function of n.
Overall, the correct statement is that g is a dependent variable, n is an independent variable, and there is a functional relationship between g and n.
g is directly proportional to n
For the function y=tan theta, which angle measure has the same value as 2pi/3 ?
A.5pi/3
B. 7pi/6
C. 4pi/3
D. pi/6
First, we need to find the value of `tan(2pi/3)`.
We know that `tan(theta) = opposite / adjacent`.
For a 30-60-90 triangle (with angles in degrees), the opposite side is `sqrt(3)` times shorter than the hypotenuse, and the adjacent side is half the length of the hypotenuse.
Therefore, in a 30-60-90 triangle with hypotenuse of length 1, the opposite and adjacent sides have lengths `sqrt(3)/2` and `1/2`, respectively.
Since `2pi/3` is equivalent to `120` degrees, we can create a triangle with one angle of `120` degrees and use the ratios above to find the value of `tan(2pi/3)`:
```
|
| /\
| / \
| /____\
|/ 1
|-------
1/2
```
From this triangle, we see that `tan(2pi/3) = (sqrt(3)/2) / (1/2) = sqrt(3)`.
We now want to find the angle that also has a `tan` value of `sqrt(3)`. Taking the inverse tangent (or arctangent) of both sides, we get:
`2pi/3 = arctan(sqrt(3))`
Using a calculator, we find that `arctan(sqrt(3)) = pi/3`.
Since `pi/3` is not one of the answer choices, we need to use the relationship between tangent values for angles in different quadrants.
The tangent function has a period of `pi`, which means it repeats its values every `pi`.
Since `2pi/3` is in the second quadrant (where `theta` is between `pi/2` and `pi`), we know that the tangent function is negative.
Using the relationship between tangent values for reference angles (angles between `0` and `pi/2`), we know that `tan(pi/3) = sqrt(3)`.
Since `pi/3` is in the first quadrant (where `theta` is between `0` and `pi/2`), we know that the tangent function is positive.
Therefore, the angle measure with the same value of `tan` as `2pi/3` is the angle `pi - pi/3 = 2pi/3` (since `tan` is negative in the second quadrant and positive in the third quadrant).
So the answer is (A) 5pi/3.
From the given information, we have:
g = f(n)
This means that the amount of milk produced (g) is a function of the number of dairy cows (n), denoted by f(n).
Based on this information, we can make the following statements:
1. The amount of milk produced (g) is dependent on the number of dairy cows (n).
2. As the number of dairy cows (n) increases, the amount of milk produced (g) is expected to increase.
3. As the number of dairy cows (n) decreases, the amount of milk produced (g) is expected to decrease.
Therefore, all three statements are true about the variables.