Sin 2x - sin x=0
Using the trigonometric identity: sin 2x=2 sinx cosx
2 sinx cosx - sinx =0
Common factor sinx
sinx ( 2 cosx -1)=0
Two options:
1) sinx=0
on the interval [0,2π), the sinx=0 for x=0 and x=<span>π=3.1416→x=3.14
2) 2 cosx - 1=0
Solving for cosx
2 cosx-1+1=0+1
2 cosx = 1
Dividing by 2 both sides of the equation:
(2 cosx)/2=1/2
cosx=1/2
cosx is positive in first and fourth quadrant:
First quadrant cosx=1/2→x=cos^(-1) (1/2)→x=</span><span>π/3=3.1416/3→x=1.05
Fourth quadrant: x=</span>2π-π/3=(6π-π)/3→x=5<span>π/3=5(3.1416)/3→x=5.24
Answer: Solutions: x=0, 1.05, 3.14, and 5.24</span>
9514 1404 393
Answer:
split the number into equal pieces
Step-by-step explanation:
Assuming "splitting any number" means identifying parts that have the number as their sum, the maximum product of the parts will be found where the parts all have equal values.
We have to assume that the number being split is positive and all of the parts are positive.
<h3>2 parts</h3>
If we divide number n into parts x and (n -x), their product is the quadratic function x(n -x). The graph of this function opens downward and has zeros at x=0 and x=n. The vertex (maximum product) is halfway between the zeros, at x = (0 + n)/2 = n/2.
<h3>3 parts</h3>
Similarly, we can look at how to divide a (positive) number into 3 parts that have the largest product. Let's assume that one part is x. Then the other two parts will have a maximum product when they are equal. Their values will be (n-x)/2, and their product will be ((n -x)/2)^2. Then the product of the three numbers is ...
p = x(x^2 -2nx +n^2)/4 = (x^3 -2nx^2 +xn^2)/4
This will be maximized where its derivative is zero:
p' = (1/4)(3x^2 -4nx +n^2) = 0
(3x -n)(x -n) = 0 . . . . . . . . . . . . . factor
x = n/3 or n
We know that x=n will give a minimum product (0), so the maximum product is obtained when x = n/3.
<h3>more parts</h3>
A similar development can prove by induction that the parts must all be equal.
Mmm me too, and a big glass of coca cola Yummy! :)
Answer:
50 pounds of beans costs $40
Answer: A & B are the same answer --> 96, max
<u>Step-by-step explanation:</u>
Consider m is the degree of the numerator (top) and n is the degree of the denominator (bottom). Then the horizontal asymptote (H.A.) is based on the relationship between m and n:
- If m > n, then there is no H.A.
- If m = n, then y = coefficient of numerator ÷ coefficient of denominator
- If m < n, then y = 0
In the given problem, m = 1 and n = 1 so the H.A. is:
This is the maximum number of moose that the forest can sustain at one time.
