The period of [tex]f(x)[/tex] is [tex]\boxed{2\pi}[/tex].
Recall that [tex]\sin(x)[/tex] and [tex]\cos(x)[/tex] both have periods of [tex]2\pi[/tex]. This means
[tex]\sin(x + 2\pi) = \sin(x)[/tex]
[tex]\cos(x + 2\pi) = \cos(x)[/tex]
Replacing [tex]x[/tex] with [tex]3x[/tex], we have
[tex]\sin(3x + 2\pi) = \sin\left(3 \left(x + \dfrac{2\pi}3\right)\right) = \sin(3x)[/tex]
In other words, if we change [tex]x[/tex] by some multiple of [tex]\frac{2\pi}3[/tex], we end up with the same output. So [tex]\sin(3x)[/tex] has period [tex]\frac{2\pi}3[/tex].
Similarly, [tex]\cos(5x)[/tex] has a period of [tex]\frac{2\pi}5[/tex],
[tex]\cos(5x + 2\pi) = \cos\left(5 \left(x + \dfrac{2\pi}5\right)\right) = \cos(5x)[/tex]
We want to find the period [tex]p[/tex] of [tex]f(x)[/tex], such that
[tex]f(x + p) = f(x)[/tex]
[tex] \implies \sin(3x + p) + \cos(5x + p) = \sin(3x) + \cos(5x)[/tex]
On the left side, we have
[tex]\sin(3x + p) = \sin(3x + 2\pi + p - 2\pi) \\\\ ~~~~~~~~ = \sin(3x+2\pi) \cos(p-2\pi) + \cos(3x+2\pi) \sin(p-2\pi) \\\\ ~~~~~~~~ = \sin(3x) \cos(p-2\pi) + \cos(3x) \sin(p - 2\pi)[/tex]
and
[tex]\cos(5x + p) = \cos(5x + 2\pi + p - 2\pi) \\\\ ~~~~~~~~ = \cos(5x+2\pi) \cos(p-2\pi) - \sin(5x+2\pi) \sin(p-2\pi) \\\\ ~~~~~~~~ = \cos(5x) \cos(p-2\pi) - \sin(5x) \sin(p-2\pi)[/tex]
So, in terms of its period, we have
[tex]f(x) = \sin(3x) \cos(p - 2\pi) + \cos(3x) \sin(p - 2\pi) \\\\ ~~~~~~~~ ~~~~+ \cos(5x) \cos(p - 2\pi) - \sin(5x) \sin(p - 2\pi)[/tex]
and we need to find the smallest positive [tex]p[/tex] such that
[tex]\begin{cases} \cos(p - 2\pi) = 1 \\ \sin(p - 2\pi) = 0 \end{cases}[/tex]
which points to [tex]p=2\pi[/tex], since
[tex]\cos(2\pi-2\pi) = \cos(0) = 1[/tex]
[tex]\sin(2\pi - 2\pi) = \sin(0) = 0[/tex]
ALOT KF POINTS PLEASE HELP The measures of two exterior angles are given in the figure below.
Choose an equation that could be used to represent one of the exterior
angles and a logical conjecture about the exterior angles of a triangle.
An exterior angle is equal to the sum of two non-adjacent interior angle of the triangle. Option B
What is a triangle?The term triangle refers to a figure that has three sides. There are several types of triangles such as the;
Isosceles triangleScalene triangleRight angled triangleDespite all these types of triangle, the basic geometric properties of triangles can be applied to all types of triangle that we have mentioned above.
Considering the question, an equation that could be used to represent one of the exterior angles and a logical conjecture about the exterior angles of a triangle is that; m<4 = m<1 + m<2. An exterior angle is equal to the sum of two non-adjacent interior angle of the triangle. Option B
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A tank holds 3,000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes.
a) The slope of the secant lines are: q = 5, f(q) = 2,076: m = - 131.1, q = 10, f(q) = 1,299: m = - 106.8, q = 20, f(q) = 357: m = - 81.6, q = 25, f(q) = 84: m = - 68.1, q = 30, f(q) = 0: m = - 51.
b) The estimated slope of the line tangent to the curve at the point (x, y) = (15, 765) is - 94.2 gallons per minute.
How to estimate the slope of a tangent line by averaging two adjacent secant lines
In this problem we must determine the slope of several lines based on the information given by the table and using the secant line formula:
m = [f(q) - f(p)] / (q - p) (1)
If we know that a = 15 and p = 765, then the slope of the secant lines are:
q = 5, f(q) = 2,076
m = [2,076 - 765] / (5 - 15)
m = - 131.1
q = 10, f(q) = 1,299
m = [1,299 - 765] / (10 - 15)
m = - 106.8
q = 20, f(q) = 357
m = [357 - 765] / (20 - 15)
m = - 81.6
q = 25, f(q) = 84
m = [84 - 765] / (25 - 15)
m = - 68.1
q = 30, f(q) = 0
m = [0 - 765] / (30 - 15)
m = - 51
The slope of the line tangent to the curve at the point (x, y) = (15, 765) can be estimated by averaging the slopes of the closest secant lines:
m' = [(- 106.8) + (- 81.6)] / 2
m' = - 94.2
The estimated slope of the line tangent to the curve at the point (x, y) = (15, 765) is - 94.2 gallons per minute.
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flying against the jet stream, a jet travels 3000 in 3 hours. Flying with the jet-stream, the same jet travels 5360 miles in 4 hours. What is the rate of the jet in still air and what is the rate of the jet stream?
Based on the time taken by the jet when flying with the wind versus against it, the rate of the jet and jet streams are:
In still air = 1,170 mphRate of wind = 170 mphWhat is the speed of the jet?The speed of the jet flying against the wind is:
= 3,000 / 3
= 1,000 mph
The speed with the jet stream is:
= 5,360 / 4
= 1,340 mph
Assuming the speed of the jet is x and the rate of wind is y, the formulas are:
x - y = 1,000
x + y = 1,340
Solving gives:
2x = 2,340
x = 1,170 mph
The rate of jet stream is:
x - y = 1,000
1,170 - y = 1,000
y = 170 mph
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What number should come next in the sequence? 0.8, 0.35, -0.1, -0.55, ... FASTTTTT
Answer:
-1
Step-by-step explanation:
0.35 - 0.8 = -0.45
-0.1 - 0.35 = -0.45
-0.55 - 0.45 = -1
at a particular high school, 42% of the students participate in sports and 25% of the students participate in drama. if 53% of the students participate in either sports or drama, what is the probability that a student participates in both sports and drama?
P( A Student participates in Both sports and Drama) =0.07
What is the probability?The probability of an occurrence can be determined using the probability formula by simply dividing the favorable number of possibilities by the entire number of possible outcomes.
To find the probability that a student participates in both sports and drama:
Let
A =Student participate in Sport
B = Student participate in Drama
Thus we have:
P(A) = 0.42
P(B) = 0.25
and
P(Ac and B or A and Bc) =0.53
P(Ac and B ) + P( A and Bc) =0.53
P(B) - P(A and B) + P(A) - P(A and B) = 0.53
0.25 - P(A and B ) + 0.42 - P(A and B) = 0.53
0.25+0.42 - 2 * P( A and B) = 0.53
0.67 - 2 * P(A and B) = 0.53
0.67 - 0.53 = 2* P(A and B)
0.14 = 2 * P(A and B)
0.14 / 2 = P( A and B)
0.07 = P( A and B)
that is:
P( A and B) = 0.07
P( A Student participate in Both sports and Drama) =0.07.
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The probability that a student participates in both sports and drama is [tex]0.14[/tex].
What is the formula for P(AUB), where A and B are any two events?If [tex]A[/tex] and [tex]B[/tex] are any two events, then the probability of the joint event [tex](A\cup B)[/tex] is given by the following formula: [tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)[/tex]
Given that 42% of the students participate in sports and 25% of the students participate in drama and 53% of the students participate in either sports or drama.
Suppose [tex]S[/tex] denotes that "a student participates in sports" and [tex]D[/tex] denotes that "a student participates in drama".
So, we have [tex]P(S)=\frac{42}{100}=0.42[/tex], [tex]P(D)=\frac{25}{100}=0.25[/tex], [tex]P(S\cup D)=\frac{53}{100}=0.53[/tex].
We want to find the probability that a student participates in both sports and drama i.e., we want to find [tex]P(S\cap D)[/tex].
By the above formula, we obtain:
[tex]P(S\cup D)=P(S)+P(D)-P(S\cap D)\\\Longrightarrow P(S\cap D)=P(S)+P(D)-P(S\cup D)\\\Longrightarrow P(S\cap D)=0.42+0.25-0.53\\\therefore P(S\cap D)=0.14=\frac{14}{100}[/tex]
Therefore, the probability that a student participates in both sports and drama is [tex]0.14[/tex].
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2х+чу=4 and x + 2y = 3
Since the values on both sides are equal, hence the system of equation given has no solution
Solving simultaneous equationsSimultaneous equations are equations that consists of two or more equation and unknowns.
Given the equation below
2х+4у=4
x + 2y = 3
From equation 2;
x = 3 - 2y
Substitute into equation 1
2(3-2y) + 4y = 4
Expand
6 - 4y + 4y = 4
6 = 4
Since the values on both sides are equal, hence the system of equation given has no solution
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Any helppppppp??????
First find volume of both the cylinders:
Cylinder Volume Formula: πr²h
(r => radius, h => height)
*Container A:
r = 8
h = 15
πr²h = π(8)²(15) = 960π³
*Container B:
r = 11
h = 13
πr²h = π(11)²(13) = 1573π³
What % of container B is empty after the pumping is complete?
The water pumped from A to B, and we want to know remaining that is empty: A < B: 1573π³- 960π³ = 613π³. Now to find %:
[tex]\frac{613\pi^{3} }{1573\pi ^{3}} * 100 = 0.39 * 100 =[/tex] 39% or 40% (Nearest Tenth)
Hope it helps!
Neil emptied his change jar and noticed he only had pennies, nickels, and quarters. He had coins for a total of . He had fewer quarters than nickels. How many pennies did Neil have? The solution is
Neil had 18 dimes, 6 nickels and 10 quarters saved up in his jar making a total of 460 pennies
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
In his piggy bank, Neil has three times as many dimes as nickels and he has four more quarters than nickels. The coins total are 4.60. How many of each coin does he have.
Let x represent dimes (0.1), y represent nickels (0.5) and z represent quarter (0.25)
Hence:
0.1x + 0.05y + 0.25z = 4.6 (1)
Also:
x = 3y (2)
And:
z = y + 4 (3)
From the equations:
x = 18, y = 6, z = 10
Neil had 18 dimes, 6 nickels and 10 quarters saved up in his jar making a total of 460 pennies
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imageGiven: a = 2, b = 3, and c = 4. What is m measured angle C degree to the nearest tenth?
Given the measure of side a, b and c of the triangle, the measure of angle C to the nearest tenth is 104.5°.
What is the measured angle C?From the law of cosines;
cosC = ( a² + b² - c² ) / 2ab
Where C is the angle C and a, b, and c are the three sides of the triangle.
Given the data in the question;
Side a = 2Side b = 3Side c = 4Angle C = ?Using law of cosine;
cosC = ( a² + b² - c² ) / 2ab
We substitute the values into the equation.
cosC = ( 2² + 3² - 4² ) / ( 2 × 2 × 3 )
cosC = ( 4 + 9 - 16 ) / ( 12 )
cosC = -3 / 12
cosC = -0.25
We find the inverse of cosine.
C = cos⁻¹( -0.25 )
C = 104.477°
C = 104.5°
Given the measure of side a, b and c of the triangle, the measure of angle C to the nearest tenth is 104.5°.
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A data set has a mean of x = 75 and a standard deviation of 11.4. Find the z-score for each of the following.
Part (a)
[tex]z=\frac{85-75}{11.4} \approx \boxed{0.88}[/tex]
By similar logic, the answers to the other parts are
(b) 1.75
(c) -2.19
(d) 0
You need to haul a load of patio bricks to a job site. Each brick weighs 4 pounds 14 ounces. Your truck can carry a 3/4 -ton load. How many bricks can your truck carry in a full load?
A.
300
B.
307
C.
362
D.
409
E.
483
Write 2.154 × 10∧5 in decimal form.
The decimal notation for the given value in scientific notation is 215400.
How to convert from scientific to decimal notation?Very large or very small numbers are denoted in scientific notation to simplify complex calculations.
The scientific notation is represented by: a × 10ⁿ
Wher a - any number or decimal number
n - the power of 10; it is an integer number
To convert this into decimal notation,
If the value of n is positive, then the decimal point in the 'a' moves right. (OR)If the value of n is negative, then the decimal point in the 'a' moves left.Calculation:The given number is in the scientific notation,
2.154 × 10⁵
Since the power of 10 is positive, the decimal point in the number 2.154 moves towards the right. I.e,
2.154 × 10⁵ = 215400
Here power is 5. So, the decimal point moved five places to the right.
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According to indical equation solve 3×=1\81
Answer:
x = - 4
Step-by-step explanation:
using the rule of indices
[tex]\frac{1}{a^{n} }[/tex] = [tex]a^{-n}[/tex] , then
[tex]3^{x}[/tex] = [tex]\frac{1}{81}[/tex] = [tex]\frac{1}{3^{4} }[/tex] = [tex]3^{-4}[/tex] , that is
[tex]3^{x}[/tex] = [tex]3^{-4}[/tex] , thus
x = - 4
Ms. Oshiro teaches her 27 students to make folded-paper birds. Each student can make about 2 birds in 1 minute. About how many birds can all the students make in 55 minutes? A 3,000 B 1,500 C 100 D 60
Answer:
The answer is 2970
Step-by-step explanation:
Collectively, all the students can make about 54 books rds per minute (27 * 2)
Therefore in 55 minutes there should be 2970
(54*55) :)
which of the following could be the ratio between the lengths of the two legs of a 30-60-90 triangle
Answer:
A, C
Step-by-step explanation:
See attached image.
(1 point) Find y as a function of x if
y" - 12y" + 32y' = 0,
y(0) = 5, y'(0) = 2, y" (0) = 1.
0
y(x) =
GEOMETRY
Surface area of a cube or a rectangular prism
There is a rectangular prism with a length of 8 yd, a width of 6 yd, and a height of 5 yd.
Find the surface area of this rectangular prism.
Be sure to include the correct unit in your answer.
Barbara runs once around the track in 3 minutes 45 seconds. Liad runs around the track in 2
minutes 30 seconds. At these rates, after 10 minutes of running, the distance Barbara has run is
what fractional part of Liad's?
Answer:
2/3rds of liad's run
Step-by-step explanation:
3 min 45s = 3 3/4 min = 15/4 min; 10/(15/4) = 8/3 laps
2min 30s = 2 1/2 min = 5/2 min; 10/(5/2) = 4 laps
(8/3)/4 = 2/3
One circle has a circumference of 12x cm. Another circle has a circumference of 32x cm. What is the ratio of the
radius of the smaller circle to the radius of the larger circle?
O 3:8
O 8:3
O 1:3
O 3:1
Answer:
3 :8
Step-by-step explanation:
12 : 32 = 3:8
I believe that is right.
here is another one that i need help on What number represents the same amount as 8 hundreds + 10 tens + 0 ones
Answer:
810 is the number that represents the same amount.
Hope this helps :)
4. Simple Interest: An investment earned 4% simple interest for 8 years. At its maturity, it was worth $5000. What amount was invested? (3
Answer:
To solve the problem we have to take 4 percent of 5000 and multiply it by 8 years and the final result is 1600 dollars, which would be the final result.
b) The diameter of a circular coin is 1.14 cm. How many coins of the same size are required to place in a straight row to cover 2.85 m length (without leaving any gap between each coin)
Answer:
250.
Step-by-step explanation:
if one coin covers 1.14 cm, then the required number can be calculated according the equality:
required_number=(given_length/given_diameter);
Then:
[tex]number=\frac{2.85*100}{1.14} =\frac{285}{1.14} =250[coins].[/tex]
P.S. note, the given length is in [meter], the diameter is in [cm].
ill give you brainliest :)
Following the instructions for the inequalities gives the following results:
1. 4<7, both sides multiplied by 7
= 28<49, both sides multiplied by 6
= 168<294, both sides multiplied by 3
= 504<882, both sides multiplied by 10
= 5,040<8,820
2. 11>2, add 5 to both sides
= 16>3, add 3 to both sides
= 19>9, add -4
= 15>5
3. -4<-2, subtract 6
= -10<-8, subtract 8
= -18<-16, subtract 2
= -20<-18
4. -8<8, divide by -4
= 2>-2, divide by -2
= -1<1
5. The inequalities remain correct at the end because the same mathematical operations are performed on both sides simultaneously.
What are inequalities?Inequality is a relationship of a non-equal comparison between two numbers or algebraic expressions.
Inequality can be represented as greater than, greater than or equal to, less than, or less than or equal to.
Thus, the rules for inequalities show that if one adds to, subtracts from, multiplies by, or divides by the same number on both sides of an inequality, the inequality remains true.
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Solve:
55) A pool is filling at 3 pts per min. How many gallons per hour is that?
What is the value of the last digit in 3,500.45
Answer:
five-hundredths
Step-by-step explanation:
A bookstore carries 48 magazines. 75% of the magazines are women's magazines. How many women's magazines does the bookstore carry?
Answer:
36
Step-by-step explanation:
75/100
75/100=0.75
0.75*48 =36
The function y = x2 + 3 is a transformation of the function y = x2. How is the y-intercept of the function affected by the transformation?
The y-intercept of the graph of y = x2 is shifted down 3 units.
The y-intercept of the graph of y = x2 is shifted to the left 3 units.
The y-intercept of the graph of y = x2 is shifted to the right 3 units.
The y-intercept of the graph of y = x2 is shifted up 3 units.
-----------------------------------------------------------------------------------------------
Answer on Edmentum is:
The y-intercept of the graph of y = x2 is shifted up 3 units.
Explanation:
The graph of a function in the form y = f(x) + k is the graph of f shifted up k units. The graph of y = x2 + 3 is the graph of y = x2 shifted 3 units up, so the y-intercept of the graph of y = x2 is shifted up 3 units.
The y-intercept of the graph of y = x2 is shifted up 3 units.
Exponential equationThe standard exponential equation is expressed as y = ab^x. Exponential equation is inverse of logarithmic equation.
Translation of coordinatesTranslations is a transformation technique that changes the position of an object from one point on the plane to another.
Given the parent function f(x) = x^2
If the function is shifted 3 units up, the function g(x) is derived by adding 3 to the function f(x). The resulting function is given as;
g(x) = f(x) + 3
g(x) = x^2 + 3
Hence the equation of the new graph g(x) is expressed as g(x) = x^2 + 3
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If 5 bags weigh the same as 3 blocks, how many blocks will 9 bags weigh?
OA) 3 blocks
OB) 4.4 blocks
OC) 5.4 blocks
OD) 15 blocks
Answer: C. 5.4 Blocks
Step-by-step explanation:
Given information
5 bags = 3 blocks
Set variable
Let x be the number of blocks
Constructure proportional equation
[tex]\frac{3}{5} } ~=~\frac{x}{9}[/tex]
Cross multiply the fraction
[tex](3)~*~(9)~=~(5)~*~(x)[/tex]
[tex]27~=~5x[/tex]
Divide 5 on both sides
[tex]27~/~5~=~5x~/~5[/tex]
[tex]\Large\boxed{x=5.4~blocks}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Answer:
oc
Step-by-step explanation:
5 times .6 equals 3
9 times .6 equals 5.4
Write 80.54 as a mixed number.
Answer:
Step-by-step explanation:
[tex]80.54=80+0.54=80+\frac{54}{100} =80+\frac{27}{50} =80\frac{27}{50}[/tex]
The question is attached
Answer:
We call the denominator rational.
Step-by-step explanation:
For example:
[tex]\frac{1}{\sqrt{2}} = \frac{\sqrt2}{\sqrt2 \cdot \sqrt2} = \frac{\sqrt2}{2}[/tex]
It's clear that the denominator is now a rational number, not an irrational one anymore.
