[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$2100\\ r=rate\to 2\%\to \frac{2}{100}\dotfill &0.02\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &5 \end{cases} \\\\\\ A = 2100\left(1+\frac{0.02}{1}\right)^{1\cdot 5}\implies A=2100(1.02)^5 \implies A \approx 2318.57[/tex]
Please help me describe how to simplify this expression.
we have successfully simplified the expression x⁵-x³ into x³(x + 1)(x - 1). This can be useful for further algebraic manipulation or simplification in larger expressions.
How to simplify?
To simplify the expression x⁵-x³, we can factor it using the distributive property of multiplication. First, we can factor out x³ from both terms:
x⁵ - x³ = x³(x² - 1)
Now, we can further simplify by recognizing that x² - 1 is a difference of squares, which can be factored into (x + 1)(x - 1). So, we have:
x⁵ - x³ = x³(x² - 1) = x³(x + 1)(x - 1)
This is the fully simplified form of the expression. We can also expand the expression to verify that it is equivalent to the original expression:
x³(x + 1)(x - 1) = (x³ * x) + (x³ * -1) + (x³ * 1) + (x³ * -1) = x⁴ - x³ + x³ - x³ = x⁴ - x³
So, we have successfully simplified the expression x⁵-x³ into x³(x + 1)(x - 1). This can be useful for further algebraic manipulation or simplification in larger expressions.
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5. A child has eight cans of soft drinks,
consisting of four different brands
with one regular and one diet drink
for each brand. The child arranges four of the cans
in a row in a random manner. Find the probability
that the
(a) arrangement consists of all diet drinks.
(b) first two are diet drinks and the second two are
regular.
The probability of choosing all diet drinks is then: P(all diet drinks) = 24 / 70 ≈ 0.343 or about 34.3%. P(first two are diet drinks and second two are regular) = 288 / 70 ≈ 4.114 or about 4.1%
What is probability?It is expressed as a number between 0 and 1, where 0 represents an impossible event (i.e., it will never happen), and 1 represents a certain event (i.e., it will always happen).
According to question:There are a total of 8 cans of soft drinks, and the child chooses 4 cans to arrange in a row. The total number of ways to choose 4 cans out of 8 is:
C(8,4) = 8! / (4! * 4!) = 70
(a) To find the probability that all four cans are diet drinks, we need to count the number of ways to choose 4 cans that are all diet drinks. Since there is one diet drink for each brand, there are 4 choices for the first can, 3 choices for the second can, 2 choices for the third can, and 1 choice for the fourth can. Therefore, the number of ways to choose 4 cans that are all diet drinks is:
4 * 3 * 2 * 1 = 24
The probability of choosing all diet drinks is then:
P(all diet drinks) = 24 / 70 ≈ 0.343 or about 34.3%
(b) To find the probability that the first two cans are diet drinks and the second two cans are regular drinks, we can count the number of ways to choose 2 diet drinks out of 4 and the number of ways to choose 2 regular drinks out of 4, and then multiply these two numbers together. There are:
C(4,2) = 4! / (2! * 2!) = 6
ways to choose 2 diet drinks out of 4, and also 6 ways to choose 2 regular drinks out of 4. Once we have chosen the 2 diet drinks and 2 regular drinks, there are 4! ways to arrange them in a row. Therefore, the total number of arrangements that have the first two cans as diet drinks and the second two cans as regular drinks is:
6 * 6 * 4! = 288
The probability of this arrangement is then:
P(first two are diet drinks and second two are regular) = 288 / 70 ≈ 4.114 or about 4.1%
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70 adults with gum disease were asked the number of
times per week they used to floss before their
diagnoses. The (incomplete) results are shown below:
(frequency of 6, relative frequency of 4, and cumulative frequency of 5 are blank)
# of
times
floss
per
week
0
1
2
3
4
5
Frequency
6
7
4
11
15
11
6
7
Relative
Frequency
6
0.1571
0.2143
0.1571
0.0857
0.1
Cumulative
Frequency
0.1429
64
0.0857
70
a. Complete the table (Use 4 decimal places when
applicable)
4
15
30
41
47
b. What is the cumulative relative frequency for
flossing 3 times per week?
%
Answer:
a. To complete the table:
# of times floss per week Frequency Relative Frequency Cumulative Frequency
0 6 0.0857 6
1 7 0.1 13
2 4 0.0571 17
3 11 0.1571 28
4 15 0.2143 43
5 11 0.1571 54
6 6 0.0857 60
7 7 0.1 67
Step-by-step explanation:
b. To find the cumulative relative frequency for flossing 3 times per week, we need to add up the relative frequencies for all values of flossing less than or equal to 3.
Cumulative relative frequency for flossing 3 times per week:
= relative frequency for flossing 0 times per week + relative frequency for flossing 1 time per week + relative frequency for flossing 2 times per week + relative frequency for flossing 3 times per week
= 0.0857 + 0.1 + 0.0571 + 0.1571
= 0.4
Therefore, the cumulative relative frequency for flossing 3 times per week is 40%.
what is the length of AB
the length of side AB of triangle ABC is 6 units.
How to find the length?
To find the length of side AB of triangle ABC, we can use the distance formula which is given by:
distance = sqrt((x2 - x1)² + (y2 - y1)²)
Where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the line segment.
Using the distance formula, we can calculate the length of side AB as follows:
AB = sqrt((2 - (-4))²+ (3 - 3)²)
AB = sqrt((6)²+ (0)²)
AB = sqrt(36)
AB = 6
Therefore, the length of side AB of triangle ABC is 6 units.
We can also visualize the triangle on the graph and see that side AB is a horizontal line segment between the points (-4,3) and (2,3), which have the same y-coordinate. This makes the length of AB equal to the horizontal distance between these two points, which is simply 2 - (-4) = 6.
It is important to note that the distance formula can be applied to find the length of any line segment in the coordinate plane, as long as we know the coordinates of its endpoints.
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Solve the system by substitution if you get a decimal round to the nearest hundred
Answer:
(-1.29, 8.29) or (9.29,-2.29)
Step-by-step explanation:
Find what y equals in terms of x for both equations.
Set those two equations equal to each other (Because y=y)
Solve for x (you may need to use the quadratic formula for this step)
Plug the value(s) of x into the original equation to find y.
given: MATH is a parallelogram
prove: MT bisects AH
Answer:
Step-by-step explanation:
In a parallelogram, opposite sides are parallel and equal.
MA // HT & AH is transversal.
∠MAG ≅ ∠GHT {Alternate interior angles are equal} --Angle
MA = HT {MATH is a parallelogram} ---> Side
MA // HT & MT is transversal.
∠AMG ≅ ∠HTG {Alternate interior angles area equal} ---->Angle
ΔMGA ≅ ΔTGH { Angle side angle congruent}
AG = HG {CPCT}
MT bisects AH
Find the surface area of the pyramid. The side lengths of the base are equal.
10 m
6.9 m
8m
Answer:
147.6 m²
Step-by-step explanation:
You want the surface area of a triangular pyramid with base sides of length 8 m and a slant height of 10 m. (The altitude of the base triangle is approximately 6.9 m.)
Surface areaThe surface area is the sum of the areas of the triangular faces.
SA = 1/2(8 m)(6.9 m) + 3 × 1/2(8 m)(10 m)
= 1/2(8 m)(6.9 m + 3×10 m) = (4)(36.9) m² = 147.6 m²
The surface area of the pyramid is 147.6 square meters.
23. 3 kg आलु र 2 kg प्याजको जम्मा मुल्य रु 240 आलुको दरमा 20% से वृद्धि र प्याजको दरमा 10% से कमी हुँदा 5kg आलु र 7 kg प्याजको जम्मा मुल्य रु 618 पर्न आउँछ भने 1 kg प्याजको मुल्य, 1 kg आलुको मुल्यभन्दा कति प्रतिशतले बढी वा कम हुन्छ ? पत्ता लगाउनुहोस् | The toptal cost of 3kg potato and 2 kg onion is Rs. 240. If the rate of potato increases by 20% and the rate of onion decreases by 10%, the total cost of 5 kg potato and 7 kg onion will be Rs. 618. By what percent the cost of 1 kg onion is more
Step-by-step explanation:
Here , the eqn is given as,
Let 1 kg onion cost Rs. y and 1kg potato cost Rs. y then,
3x + 2y= 240....eqn i
The When rate of potato increases by 20 percent,
New rate of unit kg potato becomes,
[tex]x \: + \: 20\% \: of \: x[/tex]
i.e. Rs.6x/5
And when price of onion decreases by 10% ,
New unit price is ,
[tex]y - 10\% \: of \: y[/tex]
i.e. Rs.9y/10
Now the second eqn becomes ,
20x + 21y= 2060
By using elimination method we get x= Rs 40 and y= Rs.60
Finally, we have to find percent change i.e
{(cost of 1kg onion - cost of 1kg potato)/ cost of 1kg potato}x 100%
This gives the ans as 50%
what is the vertex of this equation
x^2+4x-7
The money spent, M , purchasing burritos at a particular fast food restaurant varies directly with the number of burritos purchased, B . When 4 burritos are purchased, $18 is spent. How much money is spent if 20 burritos are purchased?
The function g is related to one of the parent functions
g(x) = −(x − 2)^3
a.) Identify the parent function f.
b.) Use function notation to write g in terms of f.
a. f(x) = x³
b. g(x) = -f(x-2)
Answer:
a) parent: f(x) = x³
b) g(x) = -f(x -2)
Step-by-step explanation:
You want the parent function and the given function written in terms of the parent function for g(x) = -(x -2)³.
a) Parent functionThe parent function is the function that remains after scale factors and translations are removed. In general, a transformed function will look like ...
g(x) = c·f((x-a)/b) +d
which scales the function f(x) horizontally by a factor of b, vertically by a factor of c, and translates the result by (a, d).
Here, we recognize that ...
g(x) = -(x -2)³
has c = -1, a = 2, b = 1, d = 0
and the parent function is ...
f(x) = x³
b) g in terms of fUsing the values for a, b, c, d that we recognized above, we have ...
g(x) = -f(x -2)
__
Additional comment
When scale factors are negative, they reflect the function over the relevant axis. Here, the negative vertical scale factor reflects the function vertically over the x-axis. The translation is 2 units to the right.
The volume of this cube is 64 cubic meters. What is the value of y?
y =
meters
Answer:
y = 4 meters
Step-by-step explanation:
to find a volume of a cube you do (y × y) y
so if Y equals 4 then it would look like this:
(4 × 4)4 which = 64
A binomial experiment has given number of trails n and the given success probability p.
N=2, p=0.2
A binomial experiment has given number of trails n and the given success probability n=2, p=0.2 then P(0) = 1.
What is binomial experiment?A binomial experiment is an experiment where there have a fixed number of independent trials with only have two outcomes either success or failure.
A binomial experiment has given number of trails n and the given success probability p.
n=2, p=0.2
The probability of obtaining x successes in n independent trials of a binomial experiment, where the probability of success is p, is given by binomial probability distribution P(x)= ₙ c ₓ p ˣ (1-p)ˣ
Where x = 0, 1, 2, ---- , n
so, P(0)= ₂C₀ (0.2)⁰ (1-0.2)⁰
= 1×1×1
=1
Hence P(0)= 1
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Complete question:
Maria's grandmother gave her a gift of $2,000 for her birthday in November 2015. She deposited this into a savings account on January 2, 2016, after the holidays. The savings bank offered 0.725% interest at the time. Over the next year, the inflation rate averaged 1.3%. Now consider the following statements:
Statement I: In 2016, the buying power of Maria's gift decreased by about 0.6%.
Statement II: Maria's gift earned around $14.50 in interest during 2016, but this amount did not keep up with inflation.
Statement III: Maria's savings account had less than $2000 in it by the end of 2016.
Statements I and II are correct, but not Statement III.
None of the statements are correct.
All the statements are correct.
Statements I and III are correct, but not Statements II.
Statements II and III are correct, but not Statements I.
According to the given question we can conclude that statement I and statement II are correct but not statement III.
Define interest?Interest is the cost incurred when lending and borrowing a specific amount of money from a financial standpoint. It is crucial to note that interest is calculated as a percentage.
given,
Statement I is correct because the inflation rate averaged 1.3%, and Maria's savings account earned interest at a rate of 0.725%, which is lower than the inflation rate. This means that the buying power of her gift decreased by approximately 1.3% - 0.725% = 0.575% or 0.6% (rounded).
Statement II is also correct because Maria's savings account earned interest on the deposit of $2,000 for one year at a rate of 0.725%. The interest earned can be calculated as:
Interest = Principal x Rate x Time
Interest = $2,000 x 0.00725 x 1 year
Interest = $14.50
However, statement III is not correct. Since Maria deposited the entire gift of $2,000 into the savings account and did not make any withdrawals during the year, the balance in the account at the end of 2016 would be the sum of the initial deposit and the interest earned, which is $2,000 + $14.50 = $2,014.50. Therefore, the correct option is:
Statement III is incorrect, however Statements I as well as II are true.
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Marcus is reading a 80-page book. He read the first 20 pages in 30 minutes. If Marcus continues to read a the same rate, how long will it take him to finish the book? Set up the table to represent the information from the problem. Pages Minutes 20 30 You got it! How long will it take Marcus to read his book? Pages Minutes 20 30 80
Using division, we can find that Marcus will take 2 hours or 120 minutes to read the book.
Define division?Repetitive subtraction is the process of division. It is the multiplication operation's opposite. It is described as the process of creating equitable groups. By dividing numbers, we divide a larger number down into smaller ones such that the larger number obtained will be equal to the multiplication of the smaller numbers.
Marcus is reading a book with 80 pages.
He reads first 20 pages in 30 mins.
So, to read 1 page time taken
= 30/20
= 3/2
= 1.5 mins
Now to read 80 pages, time taken will be:
80 × 1.5
= 120 mins.
Therefore, Marcus will take 2 hours or 120 minutes to read the book.
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Camila returns to work part time after
her accident. She could have
maintained all of her benefits without
working. What do you think this says
about Camila and her personal
investment in her human capital?
Camila's decision to return to work part-time instead of relying solely on her benefits reflects her personal investment in her human capital. By continuing to work, she is not only earning a salary but also investing in herself and her future career prospects.
How to solve the problem?
Camila's decision to return to work part-time after her accident instead of maintaining all her benefits without working shows that she values her human capital and is invested in it. Human capital refers to the skills, knowledge, and abilities that individuals possess, which are gained through education, training, and experience. By choosing to work, Camila is not only earning a salary, but she is also developing and enhancing her skills, knowledge, and abilities, thereby increasing her human capital.
Furthermore, Camila's decision to return to work part-time also shows that she has a strong work ethic and is committed to her career. She understands the importance of staying active in the workforce, even if it means working part-time, to maintain her professional network and stay up-to-date with industry developments. This commitment to her career and her willingness to work despite her physical limitations is a testament to her determination and resilience.
Overall, Camila's decision to return to work part-time instead of relying solely on her benefits reflects her personal investment in her human capital. By continuing to work, she is not only earning a salary but also investing in herself and her future career prospects.
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In parallelogram MNPQ if
m
∠
�
�
�
=
13
5
∘
m∠PQM=135
∘
find
m
∠
�
�
�
m∠MNP.
We can conclude that m∠MNP has a measure of 0 degrees in parallelogram MNPQ.
What is a parallelogram?A parallelogram is a four-sided geometric shape in which opposite sides are parallel and congruent (equal in length) to each other. It is a special case of a quadrilateral, which is any four-sided polygon.
In the given question,
Since opposite angles in a parallelogram are equal, we know that:
m∠MNP = m∠QMN
We also know that the sum of the interior angles of a triangle is 180 degrees. Therefore, we can find the measure of angle MQN as follows:
m∠MQN = 180 - m∠PQM = 180 - 135 = 45 degrees
Now, we can use the fact that the opposite angles in a parallelogram are equal to find the measure of angle QMN:
m∠QMN = m∠PQM = 135 degrees
Finally, we can use the fact that the angles in a triangle add up to 180 degrees to find the measure of angle MNP:
m∠MNP = 180 - m∠MQN - m∠QMN
= 180 - 45 - 135
= 180 - 180
= 0 degrees
Therefore, we can conclude that m∠MNP has a measure of 0 degrees. This means that the sides MN and PQ are parallel and do not intersect, so angle MNP does not exist.
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Suppose the weights of sumo wrestlers are normally distributed with a mean of 330lbs and a standard deviation of 15lbs. An up and coming competitor wants to defeat wrestlers whose weights are in the top 10%. What is the minimum weight of the sumo wrestlers at the highest weight of the league? Round your answer to the nearest whole number, if necessary.
I have to figure out how to use Excel's NORM.INV() function to solve this.
Answer: To use Excel's NORM.INV() function to solve this problem, we can follow these steps:
Determine the z-score corresponding to the top 10% of the distribution. We can use the NORM.INV() function to find this value. Since we want the top 10%, we'll use a probability of 0.9 and a mean of 330lbs and a standard deviation of 15lbs. In Excel, we can use the formula:
=NORM.INV(0.9,330,15,TRUE)
This gives us a z-score of 1.281552.
Once we have the z-score, we can use the formula for a normal distribution to find the corresponding weight. The formula is:
z = (x - μ) / σ
where z is the z-score, x is the weight we're trying to find, μ is the mean weight (330lbs), and σ is the standard deviation (15lbs).
Rearranging this formula to solve for x, we get:
x = z * σ + μ
Substituting in the values we know, we get:
x = 1.281552 * 15 + 330
This gives us a weight of approximately 349lbs.
Therefore, the minimum weight of the sumo wrestlers at the highest weight of the league is about 349lbs, rounded to the nearest whole number.
Step-by-step explanation:
Jose is a middle-aged professor who wants to retire early and start his own coaching classes. He decides to invest his savings and use the returns to fund his business venture.
That sounds like a smart plan! Before investing his savings, Jose should do some research to determine the best investment strategy for his goals.
What is research?
He will need to consider factors such as his risk tolerance, the potential return on investment, and the length of time he has to invest. Once he has identified an investment strategy that aligns with his goals, he should start investing his savings.
Over time, as his investments generate returns, he can use these returns to fund his business venture. It is important for Jose to regularly monitor his investments and adjust his strategy as needed to ensure that he is on track to achieve his goals. With careful planning and a well-executed investment strategy, Jose can achieve his goal of retiring early and starting his own coaching classes.
What is an investment ?
An investment refers to the purchase of an asset with the goal of generating income, capital appreciation, or both. The asset could be anything of value, such as stocks, bonds, real estate, commodities, or mutual funds. Investors typically make an investment with the expectation of earning a return on their investment over a period of time.
The return could be in the form of interest, dividends, or capital gains, depending on the type of investment. Investors may choose to invest in different types of assets based on their investment goals, risk tolerance, and investment horizon. The process of investing involves conducting research, analyzing market trends, and making informed decisions based on the available information.
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Complete question is: Jose is a middle-aged professor who wants to retire early and start his own coaching classes. He decides to invest his savings and use the returns to fund his business venture. Before investing his savings, Jose should do some research to determine the best investment strategy for his goals.
What scale factor was used to dilate point D? D(-1,-1)-D'(4,4)
Answer:
Step-by-step explanation:
scale factor is -4
D(-1 , -1) and if you divide either the x coordinate or the y coordinate by
4 - you get negative four which is the scale factor.
Emilie has a drawer full of color for ribbons the probability of randomly selecting a green ribbon is 84% which of the following describes the likelihood of selecting a green ribbon
When the probability of an event is high, we can say that the likelihood of that event occurring is also high. Conversely, when the probability of an event is low, the likelihood of that event occurring is also low.
What is probability?
In mathematics, probability is a measure of the likelihood that an event will occur. It is a way of quantifying the uncertainty or randomness associated with an event or set of events. The probability of an event is expressed as a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur. For example, the probability of flipping a fair coin and getting heads is 0.5, or 50%, since there are two equally likely outcomes (heads or tails) and getting heads is one of them. Probability theory is a fundamental branch of mathematics that has applications in many areas, including statistics, finance, engineering, and computer science, among others.
Here,
The likelihood of selecting a green ribbon from Emilie's drawer is high, since the probability of randomly selecting a green ribbon is 84%. This means that out of all the ribbons in the drawer, 84% of them are green.
Another way to interpret this is that if Emilie were to randomly select a ribbon from her drawer without looking, there is an 84% chance that the ribbon she selects will be green.
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Correct question is "Emilie has a drawer full of color for ribbons the probability of randomly selecting a green ribbon is 84% which describes the likelihood by the example of selecting a green ribbon".
Please help me to get my homework completed so that we don’t have to worry anymore and we can go back home tomorrow morning if we want to
Answer:
45.32
Step-by-step explanation:
uno de los catetos del triangulo rectángulo mide 77cm y la hipotenusa excede al otro cateto en 49 cm.
CALCULA LA HIPOTENUSA
La hipotenusa mide 137 cm.
Podemos utilizar el teorema de Pitágoras para resolver este problema, que establece que en un triángulo rectángulo, el cuadrado de la hipotenusa es igual a la suma de los cuadrados de los catetos.
Sea "a" la medida del otro cateto, entonces:
a² + 77² = (a+49)²
Desarrollando los cuadrados y simplificando, tenemos:
a² + 5929 = a²+ 98a + 2401
Restando a ambos lados a², obtenemos:
5929 = 98a + 2401
Restando 2401 a ambos lados, obtenemos:
3528 = 98a
Dividiendo por 98, obtenemos:
a = 36
Por lo tanto, el otro cateto mide 36 cm, y la hipotenusa se puede calcular utilizando el teorema de Pitágoras:
h² = a² + b²
h2 = 36² + 77²
h² = 12996 + 5929
h² = 18925
Tomando la raíz cuadrada en ambos lados, obtenemos:
h = 137
Por lo tanto, la hipotenusa mide 137 cm.
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Help need correct answer ASAP
Two skaters are practicing at the same time on the same rink. A coordinate grid is superimposed on the ice. One skater follows
the path y = - 2x + 14, while the other skater follows the curve y = - 2x^2+ 14x.
Find all the points where they might collide if they
are not careful.
Answer: To find the points where the two skaters might collide, we need to solve the system of equations:
y = -2x + 14 (Equation 1)
y = -2x^2 + 14x (Equation 2)
We can substitute Equation 1 into Equation 2 to eliminate y and get:
-2x + 14 = -2x^2 + 14x
Simplifying, we get:
2x^2 - 16x + 14 = 0
Dividing both sides by 2, we get:
x^2 - 8x + 7 = 0
This quadratic equation factors as:
(x - 1)(x - 7) = 0
So the possible values of x where the skaters might collide are x = 1 and x = 7.
To find the corresponding y-values, we can plug each value of x into either Equation 1 or Equation 2. For x = 1:
y = -2(1) + 14 = 12 (using Equation 1)
y = -2(1)^2 + 14(1) = 12 (using Equation 2)
So the skaters might collide at the point (1, 12).
For x = 7:
y = -2(7) + 14 = 0 (using Equation 1)
y = -2(7)^2 + 14(7) = 0 (using Equation 2)
So the skaters might collide at the point (7, 0).
Therefore, the two skaters might collide at points (1, 12) and (7, 0).
Step-by-step explanation:
Evaluate the definite integral.
[tex]\int\limits^7_0 {e^{x}sin(x) } \, dx[/tex]
To solve the integral [tex]\rm\int_0^7 e^x \sin(x) dx\\[/tex], we can use integration by parts. Let u = sin(x) and dv = [tex]\rm e^x[/tex] dx, then we have:
[tex]\begin{align} \rm\int \rm e^x \sin(x) dx &= \rm-e^x \cos(x) + \rm\int e^x \cos(x) dx \\&= \rm -e^x \cos(x) + e^x \sin(x) - \int e^x \sin(x) dx\end{align}[/tex]
Rearranging, we get:
[tex] \begin{align}2 \rm \int e^x \sin(x) dx &= \rm e^x (\sin(x) - \cos(x)) \bigg|^7_0 \\& \rm= e^7 (\sin(7) - \cos(7)) - 1\end{align}[/tex]
Dividing both sides by 2, we get:
[tex] \rm\int_0^7 e^x \sin(x) dx = \frac{e^7 (\sin(7) - \cos(7)) - 1}{2} \\ [/tex]
Therefore, the value of the integral is
[tex] \rm \boxed{ \rm\frac{e^7 (\sin(7) - \cos(7)) - 1}{2}}[/tex]
1. The quality control manager at a light-bulb factory needs to estimate the mean life of a new type of light-bulb. The population standard deviation is assumed to be 39 hours. A random sample of 30 light-bulbs shows a sample mean life of 400 hours. Construct and explain a 95% confidence interval estimate of the population mean life of the new light-bulb.
2. In a random sample of 360 men, 18, or older, 165 were married. Construct and explain a 99% confidence interval estimate of the true population proportion of married men, 18 or older.
3. A survey of first-time home buyers found that the sample mean annual income was $47,000. Assume that the survey used a sample of 26 first-time home buyers and that the sample standard deviation was $1,050. Compute and explain a 95% confidence interval estimate of the population mean.
4. For problem #1 above, what size sample would be needed to achieve a margin of error of 20 hours or less?
1. The 95% confident that the true population mean is 64.97 and 435.03 hours. 2. The 99% confident is between 0.407 and 0.500. 3. The 95% estimate of the mean is (45,424, 48,576). 4. Sample size is 91.27.
What is central limit theorem?A fundamental idea in statistics known as the central limit theorem (CLT) argues that, under specific circumstances, the sampling distribution of the mean of a random sample from any population tends to resemble a normal distribution as the sample size rises.
1. For 95% confidence interval we have:
CI = x ± z*(σ/√n)
Substituting the values:
CI = 400 ± 1.96*(39/√30) = (364.97, 435.03)
2. For 99% confidence interval we have:
CI = p-cap ± z*(√(p - cap(1- p-cap)/n))
Substituting the values:
CI = 165/360 ± 2.58*(√((165/360)*(195/360)/360)) = (0.407, 0.500)
Hence, we can be 99% confident that the true population proportion of married men, 18 or older, is between 0.407 and 0.500.
3. 95% estimate of the mean:
CI = x ± t*(s/√n)
Substituting the values:
CI = 47,000 ± 2.064*(1,050/√26) = (45,424, 48,576)
4. For sample size:
CI = 47,000 ± 2.064*(1,050/√26) = (45,424, 48,576)
Substitute the values:
n = (1.96*39 / 20)^2 = 91.27
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The first graph displays the race times for a group of runners from last year. The
second graph displays the runners' race times from this year.
Which graph has the larger range?
O The first graph has the larger range
The second graph has the larger range
Both graphs have the same range
The range cannot be determined for either graph
To determine which graph has the larger range, we need to understand what is meant by the term “range” in the context of graphs. the answer is: O The first graph has the larger range.
What are the maximum and minimum values in a dataset?In statistics, the range is defined as the difference between the maximum and minimum values in a dataset. It provides a measure of how spread out the data is, and can be used to identify the extent of variability within a dataset.
Looking at the first graph, we can see that the range of race times is from approximately 10 minutes to 25 minutes. Therefore, the range of this dataset is [tex]25 - 10 = 15[/tex] minutes.
Looking at the second graph, we can see that the range of race times is from approximately 8 minutes to 22 minutes. Therefore, the range of this dataset is [tex]22 - 8 = 14[/tex] minutes.
Since the range of the first graph is [tex]15[/tex] minutes and the range of the second graph is [tex]14[/tex] minutes, we can conclude that the first graph has the larger range.
Therefore, the answer is: O. The first graph has the larger range.
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For what numbers x, -2π≤ x ≤ 2π, does the graph of y = sec x have vertical asymptotes.?
The graph of y = sec(x) will have vertical asymptotes at these three values of x.
Define vertical asymptotesVertical asymptotes are vertical lines on a graph where a function approaches either positive or negative infinity as the independent variable approaches a certain value.
The secant function, y = sec(x), is defined as the reciprocal of the cosine function:
sec(x) = 1/cos(x)
The cosine function has vertical asymptotes at x = (2n + 1)π/2 for any integer n, since cosine is undefined at these points (division by zero). Therefore, the secant function will have vertical asymptotes at the same points, since dividing 1 by a very small number (close to zero) results in a very large number (tending towards infinity).
So, the values of x in the interval -2π ≤ x ≤ 2π that will cause the secant function to have vertical asymptotes are:
x = (2n + 1)π/2, where n is an integer
For n = -2, we have x = -5π/2, which is outside the interval, so we can ignore it.
For n = -1, we have x = -π/2
For n = 0, we have x = π/2
For n = 1, we have x = 3π/2
For n = 2, we have x = 5π/2, which is also outside the interval.
Therefore, the values of x in the interval -2π ≤ x ≤ 2π that will cause the secant function to have vertical asymptotes are:
x = -π/2, π/2, 3π/2
So, the graph of y = sec(x) will have vertical asymptotes at these three values of x.
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4. How much of a 560-mg sample of a radioactive substance is left after 102 years if the half-life of the substance is 17 years?
4.67 mg
8.75 mg
64 mg
3,584 mg
Answer:
4.67
Step-by-step explanation:
UV is tangent to ⨀T. What is
Step-by-step explanation:
if we extend the line VT to reach also the opposite side of the circle, we see that this line cuts the circle in half.
the whole arc angle of a half-circle is 360/2 = 180°.
now we can use the exterior angle rule :
the vertex angle (V) = 1/2 × difference of the angles of the intersected arcs.
the intersected arcs are
shorter = angle at T : the arc from U to the circle intersection with the line VT.
longer : the arc from U to the circle intersection with the extended line VT.
shorter + longer = 180°
longer = 180 - shorter
30 = 1/2 × ((180 - shorter) - shorter) =
= 1/2 × (180 - 2×shorter) = 90 - shorter
-60 = -shorter
shorter = 60°
the angle at T = 60°.