Hello!
This is a problem about interest rates.
Since we are not given the "[tex]n[/tex]" value, how many times this interest applies per time period, we can assume that we are most likely dealing with simple interest with an annual interest rate.
The simple interest formula is as follows,
[tex]A=P(1+rt)[/tex]
Where [tex]A[/tex] is the total amount, [tex]P[/tex] is the initial principal balance, [tex]r[/tex] is the annual interest rate, and [tex]t[/tex] is time in years.
Since we are given all this information, we can just solve after converting the interest rate of 4.9% to a decimal, which is 0.049.
[tex]A=11500(1+0.049*14)[/tex]
[tex]A=11500(1.686)[/tex]
[tex]A=19389[/tex]
So at the end of the 14th year, Shanna will have $19,389 in her account.
Hope this helps!
a pair of dice is rolled, and the number that appears uppermost on each die is observed. refer to this experiment and find the probability of the given event. (enter your answer as a fraction.)
If a pair of dice is rolled, and the number that appears uppermost on each die is observed, the probability of the sum of the numbers being either 7 or 11 is 2/9.
To find the probability of the sum of the numbers rolled on a pair of dice being either 7 or 11, we need to determine the number of favorable outcomes and the total number of possible outcomes.
There are six possible outcomes for each die, ranging from 1 to 6. Since we are rolling two dice, the total number of possible outcomes is 6 multiplied by 6, which is 36.
To calculate the number of favorable outcomes, we need to determine the combinations that result in a sum of either 7 or 11.
For the sum of 7, there are six possible combinations: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).
For the sum of 11, there are two possible combinations: (5, 6) and (6, 5).
Therefore, the number of favorable outcomes is 6 + 2 = 8.
The probability of the sum of the numbers being either 7 or 11 is given by the ratio of favorable outcomes to the total number of outcomes:
P(sum is 7 or 11) = favorable outcomes / total outcomes = 8/36 = 2/9.
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Complete question is:
A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. (Enter your answer as a fraction.)
The sum of the numbers is either 7 or 11.
Current Attempt in Progress The following table lists the monthly incomes (in hundreds of dollars) and the monthly rents paid (in hundreds of dollars) by a sample of six families. Monthly Income Monthly Rent 24 7.0 16 4.5 19 6.5 31 12.0 10 4.5 27 8.5 What is the 99% confidence interval for the mean monthly rent of all families with a monthly income of $2500, rounded to the nearest penny?
The 99% confidence interval for the mean monthly rent of all families with a monthly income of $2500 is given as follows:
(5.96, 22.38)
What is a t-distribution confidence interval?The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the equation presented as follows:
[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]
The variables of the equation are listed as follows:
[tex]\overline{x}[/tex] is the sample mean.t is the critical value.n is the sample size.s is the standard deviation for the sample.The critical value, using a t-distribution calculator, for a two-tailed 99% confidence interval, with 12 - 1 = 11 df, is t = 3.1058.
The parameters are given as follows:
[tex]\overline{x} = 14.17, s = 9.16, n = 12[/tex]
The lower bound of the interval is given as follows:
[tex]14.17 - 3.1058 \times \frac{9.16}{\sqrt{12}} = 5.96[/tex]
The upper bound of the interval is given as follows:
[tex]14.17 + 3.1058 \times \frac{9.16}{\sqrt{12}} = 22.38[/tex]
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Please help and thanks
a right angle so 90 degrees
Determine an expression for dy/d x = y' if [1+y]²-x+y=4 10.
The integration method you must use here is
Logarithmic q_23 = 1 Implicit q_23 = 2 Product rule q_23 = 3
The simplified expression for y' = 1/q_24y + q_25
The required expression is: y' = (1/2)(1-y)/(1+y)
Given [1 + y]² - x + y = 4 10. We need to determine an expression for dy/dx = y'.
Simplification of the given expression:
[1 + y]² + y - 4 10 = x
Differentiating w.r.t x by using the chain rule, we get:
(2[1 + y])*(dy/dx) + dy/dx + 1 = 0
(dy/dx)[2(1 + y) + 1] = - 1 - [1 + y]²
(dy/dx) = [- 1 - (1 + y)²]/[2(1 + y)]
The given expression is [1+y]²-x+y=4 10. We need to determine an expression for dy/d x = y'.
Differentiating the given equation with respect to x, we get:
2(1+y).dy/dx - 1 + dy/dx = 0
dy/dx(2+2y) = 1 - y(2+dy/dx)
dy/dx(2+2y) = (1-y)(2+dy/dx)
dy/dx = (1-y)/(2+2y)
dy/dx = (1/2)(1-y)/(1+y)
Hence, the required expression is: y' = (1/2)(1-y)/(1+y)
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Maya cuts sandwiches into halves. She has 15 sandwiches. How many halves does she make
Answer:
30
Step-by-step explanation:
15 x 2
Please help me with this question mark as A branlist please ASAP ASAP please ASAP
Answer:
x=18
y=27
explanation: 4 times 3 is 12
HELPPPPPPPP 5TH GRADE MATH Select the statement that correctly describes the relationship between these two sequences: 1, 2, 3, 4, 5 and 10, 20, 30, 40, 50.
Each term in the second sequence is 10 times the corresponding term in the first sequence.
Each term in the second sequence is double the corresponding term in the first sequence.
Each term in the second sequence is 20 times the corresponding term in the first sequence.
Each term in the first sequence is double the corresponding term in the second sequence.
Answer:
Each term in the second sequence is double the corresponding term in the first sequence.
Step-by-step explanation:
pls mark as brainliest
The correct statement that describe the two sequences is:
Each term in the second sequence is 10 times the corresponding term in the first sequence.
What is an arithmetic sequence?This is a type of sequence which have common difference between each term. It is represent mathematically as:
Tₙ = a + (n – 1)d
Where
Tₙ is the nth term
a is the first term
n is the number of terms
d is the common difference
How to determine the relationship between the sequences Sequence 1: 1, 2, 3, 4, 5 Sequence 2: 10, 20, 30, 40, 50Relationship =?Relationship = sequence 2 / sequence 1
For the 1st term Sequence 1 = 1Sequence 2 = 10Relationship =?Relationship = sequence 2 / sequence 1
Sequence 2 / sequence 1 = 10 / 1
Cross multiply
Sequence 2 = 10 × sequence 1
For the 2nd term Sequence 1 = 2Sequence 2 = 20Relationship =?Relationship = sequence 2 / sequence 1
Sequence 2 / sequence 1 = 20 / 2
Sequence 2 / sequence 1 = 10 / 1
Cross multiply
Sequence 2 = 10 × sequence 1
If we continue with the pattern above, we'll discovered that
Sequence 2 = 10 × sequence 1
Thus, we can conclude that:
Each term in the second sequence is 10 times the corresponding term in the first sequence.
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This is a 3rd grade math question. I don't know how to explain it to a 9yr old
this is 3rd grade math? jesus..
Answer:
that poor 3rd grader O.O that is hard
Step-by-step explanation:
online
Is (x+5)2 equivalent to 2x2+10x +10? Explain why or
why not?
Answer:
no
Step-by-step explanation:
no because its equal to 2x+10
How many different simple random samples of size 5 can be obtained from a population whose size is 35? The number of simple random samples which can be obtained is a (Type a whole number.)
The number of different simple random samples of size 5 that can be obtained from a population of size 35 is 324,632.
To calculate the number of different simple random samples of size 5 that can be obtained from a population of size 35, we can use the combination formula. The formula for combination is given by:
C(n, r) = n! / (r! * (n-r)!)
where n is the population size and r is the sample size.
In this case, we have n = 35 (population size) and r = 5 (sample size). Plugging in these values into the formula:
C(35, 5) = 35! / (5! * (35-5)!)
Simplifying this expression:
C(35, 5) = 35! / (5! * 30!)
Calculating the factorial values:
C(35, 5) = 35 * 34 * 33 * 32 * 31 / (5 * 4 * 3 * 2 * 1)
C(35, 5) = 324,632
Therefore, the number of different simple random samples of size 5 that can be obtained from a population of size 35 is 324,632.
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state a,b, and the y-intercept then graoh the function on a graphing calculator ONLY PROBLEMS 1, 3, 5. WILL MARK BRAINLEST PLEASE HELP MATH ISNT MY STRONG SUBJECT
Step-by-step explanation:
y-intercept=when x is 0
So just set 0 for x in all equations
1. 2(3)^x2(3^0)
2(1)=2
(0,2)
3. -5(.5)^x-5(.5)^0
-5(1)
-5=y
(0,-5)
6(3)^x6(3)^0
6(1)=6
(0,6)
Hope that helps :)
What is the value of c^2-d^2 if c+d=7 and c-d=-2?
Answer:
- 14
Step-by-step explanation:
c² - d² ← is a difference of squares and factors as
= (c + d)(c - d) ← substitute values for factors
= 7 × - 2
= - 14
What is the measure of angle x?
Answer:
47 degrees
Step-by-step explanation:
A triangle’s degrees add up to 180, so we can use that theorem
102 + 31 + x = 180
133 + x = 180
x = 47 degrees
399.2 divided by 1\10 =??
Answer:
3,992
Step-by-step explanation:
1/10 = 0.1
399.2/0.1 = 3,992
Answer:
3990
Step-by-step explanation:
Place the decimal to a fraction:
399.2 = 399 2/10
Simplify = 399 1/5
Put in improper fraction = 1995/5
Simplify more = 399/1 or 399
So:
399 divided by 1/10 = 3990
The null hypothesis is that 30% people are unemployed in Karachi city. In a sample of 100 people, 35 are unemployed. Test the hypothesis with the alternative hypothesis is not equal to 30%. What is the p-value?
After testing hypothesis with alternative-hypothesis is not equal to 30%, the p-value is 0.278.
To test the hypothesis that proportion of unemployed people in Karachi city is not equal to 30%, we perform a two-tailed test using binomial distribution.
The null-hypothesis (H₀) is that the proportion of unemployed people is 30% (p = 0.30), and the alternative-hypothesis (H₁) is that the proportion is not equal to 30% (p ≠ 0.30),
We have a sample of 100 people, and 35 of them are unemployed. we will calculate "test-statistic" "z-score" and use it to find p-value,
The "test-statistic" formula for "two-tailed" test is :
z = (p' - p₀)/√((p₀ × (1 - p₀))/n),
where p' = sample proportion, p₀ = hypothesized-proportion under the null-hypothesis, and n = sample-size,
In this case, p' = 35/100 = 0.35, p₀ = 0.30, and n = 100,
Calculating the test statistic:
z = (0.35 - 0.30)/√((0.30 × (1 - 0.30)) / 100)
= 0.05/√((0.30 * 0.70) / 100)
≈ 0.05/√(0.21 / 100)
≈ 0.05/√(0.0021)
≈ 0.05/0.0458258
≈ 1.0905
To find the p-value, we calculate probability of obtaining "test-statistic" as extreme as 1.0905 in a two-tailed test.
We know that the cumulative-probability (area) to left of 1.0905 is approximately 0.861.
Since this is a two-tailed test, we double this probability to get "p-value":
So, p-value = 2 × (1 - 0.861),
= 2 × 0.139,
= 0.278
Therefore, the p-value for hypothesis test is 0.278.
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Maya buys ice cream and onions at the store. She pays a total of $21.29. She pays a total of $3.80 for the ice cream. She buys 3 bags of onions that each cost the same amount. Write and solve an equation which can be used to determine xx, how much each bag of onions costs.
Let RP* be a complexity class defined as follows. A language L is in RP* if and only if there is a polynomial time Turing machine M and a polynomial p such that (i) if we L, then Prųe{0,1 }p(w) [M accepts (w,t)]21-(1/2)lwl and (ii) if w & L, then Prge{0,1}P(w) [M rejects (w,t)] = 1. Thus M comes "exponentially close" to deciding L with certainty. Prove that RP* = RP.
Main answer: RP* = RP.
Supporting answer:
RP* is defined as the set of languages L for which there exists a polynomial-time Turing machine M that, on input w, halts with probability at least 1/2 if w is not in L, and halts with probability at least 1 - 2-l|w| if w is in L. This definition is almost identical to the definition of RP, except that the probability of error is reduced from 1/2 to 2-l|w|. However, RP* and RP are equivalent.
To see that RP* is a subset of RP, note that if a language L is in RP*, then there exists a polynomial-time Turing machine M and a polynomial p such that (i) if we L, then Prųe{0,1 }p(w) [M accepts (w,t)]21-(1/2)lwl and (ii) if w & L, then Prge{0,1}P(w) [M rejects (w,t)] = 1. This means that M can be used as a randomized algorithm for L with error probability at most 1/2. Therefore, L is in RP.
To see that RP is a subset of RP*, note that if a language L is in RP, then there exists a polynomial-time Turing machine M and a constant c such that (i) if we L, then Prųe{0,1}c|w|[M accepts (w,t)] >= 1/2 and (ii) if w & L, then Prge{0,1}P(w) [M rejects (w,t)] < 1/2. By setting p(n) = c * 2n, we obtain a Turing machine M' that satisfies the conditions of RP*. Therefore, L is in RP*.
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Philip is recording what kind of shoes people are wearing at the mall. Out of the 12 people he has seen, 6 are wearing high heels. Considering this data, how many of the next 10 people Philip sees would you expect to be wearing high heels?
Answer: 5 people
Step-by-step explanation:
The easiest way to do this is to set up a proportion. 6/12=x/10. To solve, cross-multiply and divide. Multiply 6 and 10 (6*10=60) and divide by 12 (60/12=5).
You can also find that 6/12 is equal to 1/2 by dividing each side by 6. Then multiply 10 by 1/2 (or divide 10 by 2) to get the final answer of 5 people.
use cylindrical coordinates. find the mass and center of mass of the s solid bounded by the paraboloid z = 12x2 12y2 and the plane z = a (a > 0) if s has constant density k.
The center of mass can be determined by dividing the moment of the solid with respect to each coordinate axis by the total mass.
In cylindrical coordinates, the paraboloid and the plane can be represented as z = 12r^2 and z = a, respectively. To find the mass, we integrate the density function k over the region of the solid, which is bounded by z = 12r^2, z = a, and the region in the xy-plane where the paraboloid intersects the plane z = a. The integral becomes M = k * ∭ρ dV, where ρ is the density function.
To find the center of mass, we calculate the moments of the solid with respect to each coordinate axis. The x-coordinate of the center of mass can be obtained by dividing the moment about the x-axis by the total mass. Similarly, the y-coordinate and z-coordinate of the center of mass can be calculated by dividing the moments about the y-axis and z-axis, respectively, by the total mass.
By evaluating the triple integral and performing the necessary calculations, we can determine the mass and center of mass of the given solid in cylindrical coordinates.
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find the value of X, round to the nearest 10th
Answer:
4.0731 cm
Step-by-step explanation:
Use pythagorean Theorem which is a^2 + b^2 = c^2 where a and b are the sides of the triangle and c is the hypotenuse, in other words the longest side.
Since x is in line with the center of the circle and is perpendicular with line that measures 15.6 cm we know that one side of the triangle is half of 15.6 or 7.8.
So then we input our known values into the formula and solve for the missing one. The formula looks like this
7.8^2 + b^2 = 8.8^2
solve for b and you get about 4.07431
your very welcome
Answer:
X= 4.1
Step-by-step explanation:
Other answer wasn't rounded.
X= 4.1 cm
For a standard normal distribution, find:
P(-2.46 < z < 2.82)
Given: For a standard normal distribution, we need to find the probability between P(-2.46 < z < 2.82). It is found that P(-2.46 < z < 2.82) = 0.9905.
Explanation: Given, P(-2.46 < z < 2.82)
The standard normal distribution has a mean of μ=0 and a standard deviation of σ=1. It is called the standard normal distribution, because it is the normal distribution where z-scores correspond to the number of standard deviations above or below the mean.
A z-score tells us how many standard deviations a value is from the mean.
A positive z-score indicates a value above the mean, while a negative z-score indicates a value below the mean.
To find the probability of P(-2.46 < z < 2.82), we need to find the area under the standard normal distribution curve between -2.46 and 2.82.
To find this probability, we can use a standard normal distribution table or a calculator that has a normal distribution function.
Using a standard normal distribution table, we can find the area to the left of z=2.82 and the area to the left of z=-2.46 and then subtract the two values to find the area between these z-scores.
The area to the left of z=2.82 is 0.9974, and the area to the left of z=-2.46 is 0.0069.
Therefore, the area between these z-scores is:
P(-2.46 < z < 2.82) = 0.9974 - 0.0069
= 0.9905
Therefore, P(-2.46 < z < 2.82) = 0.9905.
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Describe the transformation from the graph (Desmos!)
Answer: (I am really not sure the answer if it is wrong I am so sorry)
h=3, k=2
Horizontal: up, 2 units
Vertical: up, 3 units
if an analyst wants to estimate the mean for an entire population (mu), the estimate would be more accurate if the analyst computed:
If an analyst wants to estimate the mean for an entire population (μ), the estimate would be more accurate if the analyst computed the sample mean.
If an analyst wants to estimate the mean for an entire population (μ), the estimate would be more accurate if the analyst computed the sample mean.
What is a population?In statistics, a population is a complete set of events that a statistician desires to investigate. A population is a collection of individuals, items, or data points that have a specific attribute of interest to the analyst.
What is an estimate? An estimate is an approximation of an unknown quantity that is dependent on imperfect or incomplete information. In statistics, an estimate is a projection of a population parameter dependent on data collected from a sample. An estimate is a numerical value generated from a statistical formula that is intended to provide an approximate value for an unknown population parameter.
What is an analyst?An analyst is an individual who examines a company or business's financial and business data to evaluate their health and determine their future development.
What does it mean to estimate the mean for an entire population?In statistics, estimating the mean for an entire population entails using a sample of data to calculate an approximate value of the mean for the entire population. The mean is the numerical value that provides information about the data set's central tendency. The analyst should choose a representative sample of the population to ensure the estimate is accurate.
What is the best way to estimate the mean of the entire population?
The estimate would be more accurate if the analyst computed the sample mean. The sample mean is an estimate of the population mean, denoted as μ. Sample mean is the average of the sampled data, and it is computed as follows;$$\overline{x}=\frac{\sum_{i=1}^{n}x_i}{n}$$
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If an analyst wants to estimate the mean for an entire population (μ), the estimate would be more accurate if the analyst computed the mean of a larger random sample.
A random sample is a method of selecting a subset from the entire population. It should be such that every member of the population has an equal chance of being chosen.
The sample should be sufficiently large and representative of the population as a whole to make reasonable inferences regarding the population. The mean value computed from a sample is used as an estimate of the true population mean. The sample mean is a random variable that can fluctuate from one sample to the next, depending on the sample that is taken. It has a standard deviation, called the standard error of the mean, that can be calculated using the following formula:standard error of the mean = standard deviation of the population / square root of the sample size.The larger the sample size, the smaller the standard error of the mean, and hence the more accurate the estimate of the population mean will be.
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tony runs 3 miles in 25 minutes. at the same rate, how many miles would he run in 20 minutes?
Answer:
2.4
Step-by-step explanation:
3 / 28 = 0.12
(we do this to see how many miles he ran in a minute)
0.12 x 20 = 2.4
(then we multiply how many miles he can run in 20 minutes)
"The life time, in tens of hours, of a certain delicate electrical component is modeled by the random variable X with probability density function:
w= c(9-x) 0 <= x <=9
0 otherwise
(a) show that ca 2 (b) Find the mean life time of a component.
(a) The integral of w(x) over its entire domain is zero, it cannot be equal to 1. This means that the given function does not satisfy the normalization property and hence cannot be a probability density function.
To show that the given probability density function is a valid one, we need to verify that it satisfies the two properties of a probability density function:
1. Non-negativity:
For 0 <= x <= 9, c(9-x) is always non-negative, since c is a positive constant and (9-x) is also non-negative in this range. For any other value of x, w(x) is zero. Hence, w(x) is non-negative for all x.
2. Normalization:
[tex]a[0,9] w(x) dx = a[0,9] c(9-x) dx[/tex]
= [tex]c a[0,9] (9-x) dx[/tex]
= [tex]c [(9x - (x^2)/2)] [from 0 to 9][/tex]
= [tex]c [(81/2) - (81/2)][/tex]
= [tex]c (0)[/tex]
=[tex]0[/tex]
(b) The given probability density function does not have a valid normalization constant and hence does not represent a valid probability distribution.
To find the mean life time of a component, we need to calculate the expected value of X using the formula:
[tex]E(X) = a[a,b] x (w(x) dx)[/tex]
where a and b are the lower and upper bounds of the domain respectively.
In this case, we have:
a = 0 and b = 9
w(x) = c(9-x)
Hence,
[tex]E(X) = a[0,9] x*c(9-x) dx[/tex]
= [tex]c a[0,9] (9x - x^2) dx[/tex]
= [tex]c [(81 x^2/2) - (x^3/3)] [from 0 to 9][/tex]
=[tex]c [(6561/2) - (729/3)][/tex]
= [tex]c (2958/3)[/tex]
To find the value of c, we can use the normalization property:
[tex]a[0,9] w(x) dx = 1[/tex]
[tex]a[0,9] c(9-x) dx = 1[/tex]
[tex]c a[0,9] (9-x) dx = 1[/tex]
[tex]c [(81/2) - (81/2)] = 1[/tex]
[tex]c * 0 = 1[/tex]
This is not possible, since c cannot be infinite.
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Which graph represents the function f(x) = 1/3 |x|?
Answer:
b
Step-by-step explanation:
For women aged 18-24. systolic blood pressures (in mm Hg) are normally distributed with a mean of 114 8 and a standard deviation of 13.1. If 23 women aged 18-24 are randomly selected, find the probability that their mean systolic blood pressure is between 119 and 122.
The probability that their mean systolic blood pressure is between 119 and 122 is 0.0807.
The given distribution is normal.
So, the formula for the standardized random variable, z can be used.
Here,Mean of the given distribution, μ = 114.8
Standard deviation of the given distribution, σ = 13.1
Number of women aged 18-24 randomly selected, n = 23
Let X be the mean systolic blood pressure of 23 randomly selected women aged 18-24.
P(X is between 119 and 122) = P((X-μ)/σ is between (119-μ)/(σ/√n) and (122-μ)/(σ/√n))
= P((X-μ)/σ is between (119-114.8)/(13.1/√23) and (122-114.8)/(13.1/√23))
= P((X-μ)/σ is between 1.35 and 2.45)
Using standard normal distribution table,P(1.35 < z < 2.45)= P(z < 2.45) - P(z < 1.35)≈ 0.9922 - 0.9115= 0.0807
Thus, the probability that the mean systolic blood pressure of the randomly selected 23 women aged 18-24 is between 119 and 122 is approximately equal to 0.0807.
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Use the GCF and distributive property to write an equivalent expression.
Answer:
2
6
9
Step-by-step explanation:
Because if you pay attention to our teacher he told us how to respond it!
Question is in picture
Answer:
27.5
Step-by-step explanation:
pythagorean theorem. a^2+b^2=c^2.
so 12^2+b^2=30^2
meaning 144+b^2=900
900-144=756
square root 756
b=27.5
Let a € (-1,1). Evaluate TT cos 20 de. 1-2a cos 0 + a2 d0
The value of the given integral is 2π [sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) (1/2) [sin(22e)/22 + sin(18e)/18] + C]
To evaluate the given expression:
∫∫ cos(20e) (1 - 2a cos(e) + a²) de dθ
We need to integrate with respect to both e and θ.
First, let's integrate with respect to e:
∫ cos(20e) (1 - 2a cos(e) + a²) de
Using the power reduction formula for cosine, we can rewrite the integrand:
= ∫ cos(20e) (1 - a² + a² cos²(e) - 2a cos(e)) de
= ∫ cos(20e) (1 - a² - a² sin²(e)) de
Now, we can integrate term by term:
= ∫ cos(20e) de - a² ∫ cos(20e) sin²(e) de - a² ∫ cos(20e) de
The first and third integrals are straightforward to evaluate:
= ∫ cos(20e) de - a² ∫ cos(20e) de - a² ∫ cos(20e) sin²(e) de
= sin(20e) - a² sin(20e) - a² ∫ cos(20e) sin²(e) de
Now, let's evaluate the remaining integral with respect to e:
= sin(20e) - a² sin(20e) - a² ∫ cos(20e) sin²(e) de
Using the trigonometric identity sin²(e) = (1 - cos(2e))/2, we can simplify the integrand:
= sin(20e) - a² sin(20e) - a² ∫ cos(20e) ((1 - cos(2e))/2) de
= sin(20e) - a² sin(20e) - (a²/2) ∫ cos(20e) - cos(20e) cos(2e) de
= sin(20e) - a² sin(20e) - (a²/2) ∫ cos(20e) de + (a²/2) ∫ cos(20e) cos(2e) de
Integrating the first term gives:
= sin(20e) - a² sin(20e) - (a²/2) ∫ cos(20e) de + (a²/2) ∫ cos(20e) cos(2e) de
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) ∫ cos(20e) cos(2e) de
Next, let's evaluate the remaining integral with respect to e:
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) ∫ cos(20e) cos(2e) de
Using the trigonometric identity cos(a) cos(b) = (1/2) [cos(a + b) + cos(a - b)], we can simplify the integrand:
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) ∫ (1/2) [cos(20e + 2e) + cos(20e - 2e)] de
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) (1/2) [∫ cos(22e) de + ∫ cos(18e) de]
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) (1/2) [sin(22e)/22 + sin(18e)/18] + C
where C is the constant of integration.
Now, we have the integral with respect to e evaluated. To find the final result, we need to integrate with respect to θ. However, we don't have any dependence on θ in the original expression. Therefore, we can simply multiply the result obtained above by 2π, as we're integrating over the entire range of θ.
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