The arc length of the curve [tex]y = ln(cos x)[/tex] from [tex]x = 0[/tex] to [tex]x = \pi /4[/tex] is [tex]\ln\left|\sqrt{2} + 1\right|[/tex]
To find the arc length of the curve[tex]y = ln(cos x)[/tex] from[tex]x = 0[/tex]to [tex]x = \pi /4[/tex], we will use the arc length formula for a curve given by y = f(x):
[tex]\text{Arc length} (L) = \int_a^b \sqrt{1 + \left[ f'(x) \right]^2} \, dx[/tex]
Where:
- a and b are the limits of integration (starting and ending points of the curve).
- f'(x) is the derivative of the function y = f(x).
First, we find the derivative of y = ln(cos x):
[tex]y = \ln(\cos x)\\y' = \dfrac{d}{dx}[\ln(\cos x)][/tex]
To find the derivative of ln(cos x), we use the chain rule:
[tex]y' &= \frac{1}{\cos{x}} \cdot (-\sin{x}) \\&= \frac{-\sin{x}}{\cos{x}}[/tex]
Now, we can find the arc length from x = 0 to x = π/4:
[tex]\text{Arc length } (L) = \int_0^{\frac{\pi}{4}} \sqrt{1 + \left( \frac{-\sin x}{\cos x} \right)^2} \, dx\text{Arc length } (L) = \int_0^{\frac{\pi}{4}} \sqrt{1 + \frac{\sin^2 x}{\cos^2 x}} \, dx\text{Arc length } (L) = \int_0^{\frac{\pi}{4}} \sqrt{\frac{\cos^2 x}{\cos^2 x} + \frac{\sin^2 x}{\cos^2 x}} \, dx\text{Arc length } (L) = \int_0^{\frac{\pi}{4}} \sqrt{\frac{\cos^2 x + \sin^2 x}{\cos^2 x}} \, dx[/tex]
Remember that [tex]cos^2 x + sin^2 x = 1[/tex], so the expression becomes:
[tex]\text{Arc length (L)} = \int_0^{\frac{\pi}{4}} \sqrt{\frac{1}{\cos^2 x}} \, dx[/tex]
Now, simplify the expression inside the square root:
[tex]\text{Arc length (L)} = \int_0^{\frac{\pi}{4}} \sqrt{\frac{1}{\cos x}} \, dx[/tex]
Integrate with respect to x:
[tex]\text{Arc length (L)} = \int_0^{\frac{\pi}{4}} \sqrt{Sec x} \, dx[/tex]
Now, evaluate the integral:
[tex]\text{Arc length} (L) = \ln\left| \sec x + \tan x \right| ~\text{from}~ 0 ~\text{to}~ \frac{\pi}{4}\text{Arc length} (L) = \ln\left| \sec \left( \frac{\pi}{4} \right) + \tan \left( \frac{\pi}{4} \right) \right| - \ln\left| \sec (0) + \tan (0) \right|[/tex]
[tex]\sec\left(\frac{\pi}{4}\right) = \sqrt{2}, \quad \tan\left(\frac{\pi}{4}\right) = 1\\\sec(0) = 1, \quad \tan(0) = 0\\\text{Arc length} (L) = \ln|\sqrt{2} + 1| - \ln|1 + 0|\\\text{Arc length} (L) = \ln|\sqrt{2} + 1|[/tex]
Using the property of natural logarithms, ln(a) - ln(b) = ln(a/b), we get:
[tex]\text{Arc length (L)} = \ln\left|\sqrt{2} + 1\right| - \ln\left|1\right|\\\text{Arc length (L)} = \ln\left|\sqrt{2} + 1\right|[/tex]
The arc length of the curve [tex]y = ln(cos x)[/tex] from [tex]x = 0[/tex] to [tex]x = \pi /4[/tex] is [tex]\ln\left|\sqrt{2} + 1\right|[/tex]
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Find the greatest common factor for 36m(r+t) and
9p(r+t) 6.
The greatest common factor for 36m(r+t) and 9p(r+t)6 is 18(r+t).
According to the question,
We have the following two expressions:
36m(r+t) and 9p(r+t)6
Now, we will first solve the second expression:
54p(r+t)
Now, the greatest common factor means the greatest number and variables which can divide both the terms completely.
So, the common variables in both the terms are (r+t).
In 36 and 54, the greatest common factor is 18 because their respective product of factors are 2*2*3*3 and 3*3*3*2.
Now, the common factors in their product are 2*3*3. So, the greatest common factor or highest common factor is 18.
Hence, the greatest common factor for 36m(r+t) and 9p(r+t)6 is 18(r+t).
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Using the graph below, determine the employee's hourly wage.
Find the domain and range for the function
f(x)={3x−1/2,4−12x,if x≤2,if x>2.
The domain and the range of the piece-wise function are given as follows:
Domain: all real values.Range: (-∞, 4].How to obtain the domain and the range of the function?The domain of a function is composed by the set that contains all possible input values of the function.
In this problem, the function is a piece-wise function, as it has two different definitions. However, for each definition, the function is a linear function, which has no input restrictions, hence the domain of the function is composed by all real values.
The range of a function is composed by the set that contains all possible output values of the function.
The outputs of the function are obtained as follows:
To the left of x = 2: increasing line, the greatest value is of -6.To the right of x = 2: decreasing line, the greatest value is of 4.Linear functions have no open spaces on it's domain, hence the interval that describes the range is given by:
(-∞, 4].
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Aiden walked a total of 6 kilometers by making 2 trips to school. How many trips will Aiden
have to make in all to walk a total of 9 kilometers? Assume the relationship is directly
proportional.
trips
Submit
Solve proportions (74)
Work it out
Not feeling ready yet? These can help:
Do the ratios form a proportion: word problems
This can be solved using proportions. Aiden needs to make 3 trips to walk 9 kilometers.
What is a proportion?
Two numerical sequences, generally experimental data, are said to be proportional or directly proportional if their corresponding elements have a constant ratio, which is known as the coefficient of proportionality or proportionality constant. Two sequences are inversely proportionate if the product of their respective elements is constant, also known as the coefficient of proportionality. This notion is frequently extended to related changing numbers, which are typically referred to as variables. For instance, if an item moves at a constant speed, the distance traveled is exactly proportional to the time spent traveling, with the speed serving as the proportionality constant.
For 2 trips, Aiden walks 6 kilometers
So, for 9 kilometers to walk he need 'x' trips (let)
Then they are in proportion, i.e.,
2/6 = x/9
or, x = (2/6)9
or, x = 3
Hence, Aiden needs 3 trips to walk a total of 9 kilometers.
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URGENT!! ILL GIVE
BRAINLIEST!!!! AND 100 POINTS!!!!!
Answer:
c. 11x + 4x = 180; x = 12
Step-by-step explanation:
If m||n, then angle 11x and angle 4x must be same-side interior angles, which means they are supplementary (adds up to 180 degrees). This means we can set up the equation
11x + 4x = 180
15x = 180
x = 12
I really need help and my paper is due at the end of the day today at 1:49pm can someone help me ???
Answer:
The pencils at store A cost more.
Step-by-step explanation:
First find the price of one pencil at store A
y=0.55(1)=
y=0.55
One pencil at store A costs 0.55
Now find the price of one pencil at store B
3.06/6=
0.51
One pencil at store B costs 0.51
0.55>0.51 so The pencils at store A cost more.
Hope this helps!
A washer and a dryer cost $889 combined. The washer costs $39 more than the dryer. What is the cost of the dryer?
Answer:
405.50
Step-by-step explanation:
hope it helped :)
Answer:
Step-by-step explanation:
405.5
To solve for the leading coefficient, use a=
Answer:
1/6
Step-by-step explanation:
I just did the problem myself! Photo evidence included!
Help me please !!!!!!!!!!!!!
The equation for the sentence below is x - 23 = 28.
The solution to the equation is x = 5.
How to model a sentence with an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Therefore, the sum of a number(x) and - 23 is 28 can be represented with an equation as follows:
The unknown number is x.
Hence,
x + (-23) = 28
Therefore, the equation from the sentence above is x - 23 = 28.
Now, let's solve the equation for x as follows;
x - 23 = 28
add 23 to both sides of the equation
x - 23 + 23 = 28 - 23
x = 5
Therefore, the value of x is 5
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Margo wants to build a rectangular fence around her garden, and she has 24 meters of wood.
Which of these dimensions will give Margo the largest area?
The dimensions which will give Margo the largest area is 6 m by 6 m
What is area?Area is defined as the total space taken up by a flat (2-D) surface or shape of an object.
Given that, Margo has 24 meters of wood to build a fence around her garden. She wants to create the largest area possible
Let the length be L and the width be W
So the perimeter equation is
2L + 2W = 24
2(L+W) = 24
L+ W= 12
L= 12-W
To find the largest area
area = L×W
area = (12-W) ×W
area= 12W- W²
A= -W²+ 12W
The vertex of the parabola is the point (h, k) where h=
= 6
to find the k part
k= (-6)² + 12(6)= 108 m²
The largest possible area is 108 m²
putting the value of W in
L= 12-W
L= 12-6 = 6 m
Hence, the length and width is same, 6 m by 6 m
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What happens when you multiply a positive number by -1? What happens when you multiply a negative number by -1?
Answer:
You get a negative answer
Step-by-step explanation:
that is literally it! you get a negative answer. hope this helps
Assume that a procedure yields a binomial distribution with
n = 5 trials and a probability of success of p=0.60. Use a
binomial probability table to find the probability that the
number of successes x is exactly 2.
Answer: what grade is this
Step-by-step explanation:
find the value of x in the following equation:
3.6(2x + 5) = 7.2x + 18
No solution
Infinite solutions
x = 0
x = 2.3
Answer:
infinite solutions
Step-by-step explanation:
both sides are identical
-(-4)-|-6| how do i solve
Answer: -2
Step-by-step explanation:
First distribute the negative sign to the negative four then find the absolute value of 6
-(-4)-|-6| = 4-6
then subtract 4 from 6 and you get -2.
P=$3000
R=10%
T=5 yrs.
I=
I need the interest answer
P=$8000
T=4 yrs.
I=$800
R=
I need the rate answer
P=$2000
R=20%
I=$400
T=
I need the time answer
The answer for the part 1 I=1500, part 2: R = 2.5% and part 3: T=1year, using Simple interest, Rate and Time.
What is simple interest and Rate?
A quick and simple way to figure out interest on money is to use the simple interest technique, which adds interest at the same rate for each time cycle and always to the initial principal amount. Any bank where we deposit our funds will pay us interest on our investment. One of the different types of interest charged by banks is simple interest. Now, before exploring the idea of basic curiosity in further detail,
The formula for simple interest is:
S = p*t*r/100
where P is the principal amount of money, R is the interest rate as a decimal, T is the time.
Given part 1: P=$3000,R=10%,T=5 yrs.
I = 3000*10*5/100 = 1500
part 2: P=$8000,T=4 yrs,I=$800
R= 800*100/8000*4 = 2.5%
part3: P=$2000,R=20%,I=$400
So T = 400*100/2000*20 = 1 year
Hence the answer for the part 1 I=1500, part 2: R = 2.5% and part 3: T=1year
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Jeff sold half of his baseball cards then bought sixteen more. He now has 21 baseball cards.
How many cards did he begin with?
2
Jeff would have had 10 cards to begin with.
Then, if Jeff sold half of his baseball cards, he would then have 5 cards.
It says that Jeff bought 16 more cards, so 16 + 5
Well, 16 + 5 = 21. The text says that he now has 21 cards, so this should be a correct answer.
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Joke: What is a snowman's favorite type of Mexican food?
Answer: Burr-itos!
Have a great day and God bless! :D
A hive of bees contains 20 bees when it is first discovered. After 5 days, there are 29 bees. It is
determined that the population of bees increases exponentially.
How many bees are will there be after 100 days? Express your answer as an exact value and rounded to the
nearest whole number.
After 100 days there will be 33750 bees in the bee hive .
in the question ,
it is given that
number of bees in the bee hive when discovered = 20 bees
number of bees after 5 days = 29 bees
also given that the population of the bees increases exponentially ,
the exponential growth is denoted by the function ,
A = A₀(eᵃˣ)
where A is the population after x days ,
A₀ is the initial population , x is the days and a is the constant of proportionality .
Substituting the values in the exponential growth formula ,
we get
29 = 20 ([tex]e^{a\times5}[/tex])
29/20 = [tex]e^{a\times5}[/tex]
taking ln both the side
ln(29/20) = ln( [tex]e^{a\times5}[/tex])
5a = ln(29/20)
a = 0.07431
to find the population after 100 days , we substitute x = 100,
we get
A = 20* [tex]e^{0.07431\times100}[/tex]
A = 33749.88
A ≈ 33750
Therefore, the number of bees in the bee hive after 100 days is 33750 .
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HELP I don’t get any of the questions.
Answer: Sir would you mind telling me what chart your using
Step-by-step explanation:
Unit 4 parallel and perpendicular lines unit test.
It is necessary to collimate the light source before using the prisim to disperse the light so that the light rays entering the prisim are parallel.
What is collimate?
Collimator is a tool for converting a point source's divergent radiation into a parallel beam. To perform specific measurements in spectroscopy, geometrical optics, and physical optics, the light must be collimated.
An optical collimator is made up of a tube with a convex lens at one end and a variable aperture at the other end that is located in the lens' focal plane. To examine the image without parallax, radiation entering the aperture exits the collimator as a parallel beam.
Therefore collimation is needed for the light rays entering the prism to be parallel.
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To find the quotient of two fractions, you first need to rewrite the division problem as an
equivalent multiplication problem. Find this quotient using multiplication.
23
4.
695 I need help asap please
Please help me with these experts i really need your responses Please and thank you
Answer:
1: 8
2: 5^4
3: 6^2 x 6^3
Step-by-step explanation:
(2^2x2)^3/(2^4x2^2)
(4x2)^3/(16x4)
8^3/64
512/64
8
2)
5^-2x5^5x5^1
5^-2+5+1
5^4
3)
6^2 x 6^3
Hopes this helps please mark brainliest
Twelve quarts of oil cost $15.00. How much do 5 qt of oil cost?
5 qt of oil cost $6.25 when twelve quarts of oil cost $15.00.
Define unitary method.The unitary approach is frequently helpful for multiplying and dividing by fractions because it is intuitive and, assuming the numbers add up, may be performed mentally. As a result, it might be a useful way to explain or introduce the often used written algorithms for fractions, especially division by a fraction. The unitary approach involves calculating the value of a single unit, from which we can calculate the values of the necessary number of units.
Given,
12 qt = 15.00
For one unit, we will use unitary method:
1 qt = 15/12
1 qt = 5/4
For 5 qt of oil:
5 qt = 5/4 × 5
5 qt = 25/4
Converting it in decimals:
5 qt = 6.25
5 qt of oil cost $6.25 when twelve quarts of oil cost $15.00.
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Pls help
first person gets brainliest
Answer:
x = -4
The x+43 angle is 39 degrees
The x+66 angle is 62 degrees
======================================================
Explanation
Rule: The three inside angles of any triangle always add to 180 degrees.
In this case we can say the following
(x+43)+(x+66)+(79) = 180
2x+188 = 180
2x = 180-188
2x = -8
x = -8/2
x = -4
That leads us to
x+43 = -4+43 = 39x+66 = -4+66 = 62The three interior angles of this triangle are
39 degrees62 degrees79 degreesThey add to 39+62+79 = 180 to confirm the answer.
what is 1/10 to the 3rd power.
Answer: 0.001
Step-by-step explanation:
(t)=61*(0.96)^t
complete the following sentence about the hourly rate of change in the medicine concetration
Every hour, the medicine concentration decays by a factor of 0.09.
What is an exponential function?An exponential function simply refers to a mathematical function whose values are generated by a constant that is raised to the power of the argument. Mathematically, an exponential function can be modeled or represented by using this equation:
f(x) = abˣ
Where:
a represents the initial value.b represents the rate of change.At the initial time (t), the medicine concentration is given by this exponential function:
C(t) = 61 × (0.96)^t
Note: One (1) hour is equal to sixty (60) minutes.
After an hour, the medicine concentration is given by this exponential function:
C(t + 60) = 61 × (0.96)^{t + 60}
Therefore, the rate of change every hour in the medicine concentration would be calculated as follows:
Decay factor = [C(t + 60)/C(t)]
Decay factor = [61 × (0.96)^{t + 60}]/61 × (0.96)^t
Decay factor = (0.96)^t
Decay factor = (0.96)^60
Decay factor = 0.0864 ≈ 0.09.
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The probability that an oven will have failed
after 1000 hours of use is 0.09.
What is the probability that it will still be
working after 1000 hours of use? Give your
answer as a decimal
Answer:
0.91
Step-by-step explanation:
1 - .09 = 0.91
x -2 0 2 3 y 0 6 12 5 The zero of the function represented in this table is -2. What do you think that means?
The zero of the function shown in this table is -2, which means that the presented function has an x-intercept.
What is x-Intercept?The x-intercept is defined as an intercept that is located at the x-axis of the plane, which is the location or coordinates from where the line crosses.
The table is given in the question,
As per the given table, the value of x is -2 then the value of y = 0. This represents the x-intercept of the function. The zero of the function represented in this table is -2 which means x-intercept in the given function.
Thus, the zero of the function shown in this table is -2, which means that the presented function has an x-intercept.
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the planet Tott, there are 10 rews to every 3 frophs. If a farmer has
50 rews on her pisquet farm, how many frophy are on the farm
Your answer may be exact or accurate to the nearest tenth.
The number of Frophy needs is 15.
What is ratio?
A ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent and the divisor or number that is dividing termed the consequent.
Given,
There are 10 rews to 3 frophy.
The ratio for one rew = 3/10 frophy
The value for 50 rew = 3(50)/10
= 15
There are 15 frophy for 50 rews.
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gabby uses 5 cups of corn and 6 cups of lima bean to make succotash. Robert uses 7 cups of corn and 9 cups of lima beans to make succotash. are the ratios of corn to lima beans equivalent? Explain why or why not
The ratios of corn to lima beans are not equivalent.
What are equivalent ratios?
The ratios that are same when compared are called equivalent ratios. The equivalentity of two or more ratios can be determined by comparing them to one another. For instance, 1:2 and 2:4 have the same ratio.
In case of Gabby:
Ratio of corn to lima beans = 5/6 = 0.833
In case of Robert:
Ratio of corn to lima beans = 7/9 = 0.777
Notice that 0.833 ≠ 0.777
So, the ratios of corn to lima beans are not equivalent.
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A farm let’s u pick 3 pints of raspberries for $12
1) what is the cost per pint
2) how many pints do you get per dollar
3) at this rate, how many pints can you afford for $20.00
4) at this rate, how much will 8 pints of raspberries cost?
The cost per pint =4
Pints do you get per dollar = 1/4
Pints can you afford for $20.00 = 5
8 pints of raspberries cost = 32
Unit rates are defined as rates that are expressed as a number of one, such as 3 feet per second or 4 miles per hour.
According to the given statement: For $12, you can pick three pints of raspberries at a farm.
then, according to the unit rate definition,
cost per unit = 12/3 = 4
Unit rate per dollar =3/12
= 1/4
= 0.25 pint
0.25 pints of raspberries get per dollar.
Pints afford for $20
= 20/4 =5
8 pints of raspberries cost
= 4 x 8
= 32
Therefore
The cost per pint =4
Pints do you get per dollar = 1/4
pints can you afford for $20.00 = 5
8 pints of raspberries cost = 32
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