Answer:
Period T of the given function f(x) = sin(2x) + cos(4x)
= π
Step-by-step explanation:
Given that y(x) is a sum of two trigonometric functions. The period T of sin 2x would be (2π÷2) = π. Period T of cos4x would be (2π÷4) that is π/2
Find the LCM of π and π/2 . That would be π. Hence the period of the given function would be π
The fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \) is:\[ T = \pi \][/tex].
To determine the fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \)[/tex], we need to find the smallest positive value of ( T ) such that [tex]f(x + T) = f(x) \) for all \( x \).\\[/tex]
Let's start by analyzing the periods of the individual components of [tex]\( f(x) \).[/tex]
1. Period of [tex]\( \sin(2x) \):[/tex]
The standard period of [tex]\( \sin(x) \) is \( 2\pi \). For \( \sin(2x) \),[/tex] the argument ( 2x ) scales the period. To find the period of [tex]\( \sin(2x) \)[/tex], we set:
[tex]\[ 2x = 2x + 2\pi \]\\[/tex]
Solving for the period, we get:
[tex]\[ x = x + \frac{2\pi}{2} \][/tex]
Thus, the period of [tex]\( \sin(2x) \)[/tex] is:
[tex]\[ \frac{2\pi}{2} = \pi \][/tex]
2. Period of [tex]\( \cos(4x) \):[/tex]
The standard period of [tex]\( \cos(x) \) is \( 2\pi \). For \( \cos(4x) \)[/tex], the argument ( 4x ) scales the period. To find the period of [tex]\( \cos(4x) \),[/tex] we set:
[tex]\[ 4x = 4x + 2\pi \][/tex]
Solving for the period, we get:
[tex]\[ x = x + \frac{2\pi}{4} \][/tex]
Thus, the period of [tex]\( \cos(4x) \)[/tex]is:
[tex]\[ \frac{2\pi}{4} = \frac{\pi}{2} \][/tex]
Next, we need to find the smallest positive ( T ) such that both [tex]\( \sin(2x) \)[/tex] and [tex]\( \cos(4x) \)[/tex] have the same period ( T ). This means that ( T ) must be a common multiple of the periods of the two components, [tex]\( \pi \)[/tex] and [tex]\( \frac{\pi}{2} \).[/tex]
To find the fundamental period ( T ), we determine the least common multiple (LCM) of [tex]\( \pi \) and \( \frac{\pi}{2} \):[/tex]
- [tex]\( \pi \)[/tex]can be written as [tex]\( \pi \times 1 \).[/tex]
- [tex]\( \frac{\pi}{2} \)[/tex] can be written as [tex]\( \pi \times \frac{1}{2} \).[/tex]
The LCM of [tex]\( 1 \) and \( \frac{1}{2} \) is \( 1 \) since \( 1 \)[/tex] is the smallest number that both [tex]\( 1 \) and \( \frac{1}{2} \)[/tex] can divide without leaving a remainder.
Thus, the LCM of [tex]\( \pi \) and \( \frac{\pi}{2} \) is:[/tex]
[tex]\[ \text{LCM}\left(\pi, \frac{\pi}{2}\right) = \pi \][/tex]
Therefore, the fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \) is:\[ T = \pi \][/tex]
Consider the relationship between the number of bids an item on eBay received and the item's selling price. The following is a sample of 5 items sold through an auction. Price in Dollars 26 29 32 38 47 Number of Bids 12 13 15 16 18 Step 3 of 3 : Calculate the correlation coefficient, r. Round your answer to three decimal places.\
Answer:
Let's assume the following data:
Price in Dollars (X) 26 29 32 38 47
Number of Bids (Y) 12 13 15 16 18
For our case we have this:
n=10 [tex] \sum x = 172, \sum y = 74, \sum xy = 2623, \sum x^2 =6194, \sum y^2 =1118[/tex]
[tex]r=\frac{5(2623)-(172)(74)}{\sqrt{[5(6194) -(172)^2][5(1118) -(74)^2]}}=0.974[/tex]
So then the correlation coefficient would be r =0.974
Step-by-step explanation:
Previous concepts
The correlation coefficient is a "statistical measure that calculates the strength of the relationship between the relative movements of two variables". It's denoted by r and its always between -1 and 1.
Solution to the problem
And in order to calculate the correlation coefficient we can use this formula:
[tex]r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}[/tex]
Let's assume the following data
Price in Dollars (X) 26 29 32 38 47
Number of Bids (Y) 12 13 15 16 18
For our case we have this:
n=10 [tex] \sum x = 172, \sum y = 74, \sum xy = 2623, \sum x^2 =6194, \sum y^2 =1118[/tex]
[tex]r=\frac{5(2623)-(172)(74)}{\sqrt{[5(6194) -(172)^2][5(1118) -(74)^2]}}=0.974[/tex]
So then the correlation coefficient would be r =0.974
The circle below represents one whole.
What percent is represented by the shaded area?
The region represented by the shaded area has 25% and accounts for one-quarter of the overall circle.
The area is the space occupied by any two-dimensional figure in a plane. The area of the circle is the space occupied by the circle in a two-dimensional plane.
The formula for calculating circle area is r2, where r is the radius of the circle.
The entire area of the circle in the accompanying figure is r2. The shaded area accounts for one-fourth of the circle's overall area.
Total area = πr²
Shaded area = ( 1 / 4 )πr²
The percentage of the shaded area will be calculated as
Shaded area = ( 1 / 4 )πr²
Shaded area = ( 0.25 )πr²
To convert it into a percentage multiply by 100.
Shaded area = ( 0.25 x 100 )πr²
Shaded area =25% πr²
Put πr² as the total area.
Shaded area =25% of Total area.
Therefore, the region represented by the shaded area has 25% and is 1/4th of the total circle.
Round your answer to the nearest hundredth. Again.
Given:
In the given triangle ΔABC,
AB = 9 unit
AC = 2 unit
To find the value of ∠ABC.
Formula
From trigonometric ratio we get,
[tex]sin \theta = \frac{opposite}{hypotenuse}[/tex]
Let us take, ∠ABC = [tex]\theta[/tex]
With respect [tex]\theta[/tex], AC is the opposite side and AB is the hypotenuse.
So,
[tex]sin \theta = \frac{AC}{AB}[/tex]
or, [tex]sin \theta[/tex] = [tex]\frac{2}{9}[/tex]
or, [tex]\theta = sin^{-1} (\frac{2}{9} )[/tex]
or, [tex]\theta= 12.84^\circ[/tex]
Hence,
The value of ∠ABC is 12.84°.
To avoid a service fee, your checking account balance must be at least $300 at the end of each month. Your current balance is $337.03. You use your debit card to spend $132.78. What possible amounts can you deposit into your account by the end of the month to avoid paying the service fee?
A deposit of at least $95.75 is needed to avoid the service fee, as this will bring the balance from $204.25 back to the required $300 minimum.
To avoid a service fee, we need to ensure that the checking account balance is at least $300 at the end of the month. Starting with a balance of $337.03 and after spending $132.78, the new balance is calculated as follows:
$337.03 - $132.78 = $204.25.
To avoid the service fee, the account balance must return to at least $300. Therefore, you need to deposit the difference between your current balance and the minimum balance required:
$300 - $204.25 = $95.75.
Any deposit amount greater than or equal to $95.75 will therefore avoid the service fee.
In a recent survey in a Statistics class, it was determined that only 74% of the students attend class on Fridays. From past data it was noted that 88% of those who went to class on Fridays pass the course, while only 20% of those who did not go to class on Fridays passed the course.
a.What percentage of students is expected to pass the course?
b.Given that a person passes the course, what is the probability that he/she attended classes on Fridays?
Answer:
a) 83%
b) 0.892
Step-by-step explanation:
percentage that attends class on friday = 74%
percentage that pass because they attend class on friday = 88%
percentage that pass but did not go to school on friday = 20%
a) percentage of students expected to pass the course
= (74% x 88%) +(88% x 20%)
= 0.6512 + 0.176
= 0.8272
= 83%
b) If a person passes the course, what is the probability that he/she attended classes on Fridays
= 74% divided by 83%
= 0.892
80% of 25 is equal to what
Answer:
20
Step-by-step explanation:
Of means multiply and is means equals
80% * 25 = ?
Change to decimal form
.80 *25 =
20
An ant moves along the x-axis from left to right at 5 inches per second. A spider moves along the y-axis from up to down at 3 inches per second. At a certain instant, the ant is 4 inches to the right of the origin and the spider is 8 inches above the origin. At this instant, what is the rate of change of the distance between the spider and the ant
Answer: The rate of change of the distance between the spider and the ant is 4.92 inches/sec
Step-by-step explanation: Please see the attachments below
Find the slope
(-19,-6) (15,16)
Answer:
11/17
Step-by-step explanation:
slope between two points: slope = (y2 - y1) / (x2 - x1)
(x1, y1) = (-19, -6), (x2, y2) = (15, 16)
m = (16 - ( - 6)) / (15 - ( - 19))
refine
m = 11/17
sorry it is hard to follow... i am on my phone rn :/
Final answer:
The slope between the points (-19, -6) and (15, 16) is 11/17.
Explanation:
To find the slope of the line connecting the points (-19,-6) and (15,16), we will use the slope formula which is the change in y-coordinates divided by the change in x-coordinates. Here is the process:
Identify the coordinates of the two points. Point 1 is (-19, -6), and Point 2 is (15, 16).Apply the slope formula: m = (y2 - y1) / (x2 - x1).Substitute the given values into the formula: m = (16 - (-6)) / (15 - (-19)) = (16 + 6) / (15 + 19).Simplify: m = 22 / 34.Reduce to the simplest form: m = 11 / 17.Therefore, the slope of the line connecting the two points is 11/17.
Standardization of a Normal Distribution: Bryce reads in the latest issue of Pigskin Roundup that the average number of rushing yards per game by NCAA Division II starting running backs is 50 with a standard deviation of 8 yards. If the number of yards per game (X) is normally distributed, what is the probability that a randomly selected running back has 64 or fewer rushing yards
Answer:
[tex]P(X<64)=P(\frac{X-\mu}{\sigma}<\frac{64-\mu}{\sigma})=P(Z<\frac{64-50}{8})=P(z<1.75)[/tex]
And we can find this probability using the normal standard table or excel and we got:
[tex]P(z<1.75)=0.9599[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the number of rushing yards of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(50,8)[/tex]
Where [tex]\mu=50[/tex] and [tex]\sigma=8[/tex]
We are interested on this probability
[tex]P(X<64)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(X<64)=P(\frac{X-\mu}{\sigma}<\frac{64-\mu}{\sigma})=P(Z<\frac{64-50}{8})=P(z<1.75)[/tex]
And we can find this probability using the normal standard table or excel and we got:
[tex]P(z<1.75)=0.9599[/tex]
Deandre just bought 9 bags of 15 cookies each. He already had 6 cookies in a jar. How many cookies does deandre have now?
Answer:
141 cookies
Step-by-step explanation:
amount of cookies
= 9(15) + 6
= 135 + 6
= 141
A professor at a local university noted that the grades of her students were normally distributed with a mean of 78 and a standard deviation of 10. The professor has informed us that 16.6 percent of her students received grades of A. What is the minimum score needed to receive a grade of A?
To find the minimum score needed to receive a grade of A, we need to determine the cutoff point for the top 16.6% of students. We can use the Z-score formula to convert a raw score into a standardized score and then find the corresponding raw score. The minimum score needed to receive a grade of A is approximately 88.
Explanation:To find the minimum score needed to receive a grade of A, we need to determine the cutoff point for the top 16.6% of students. In a normal distribution, we can use the Z-score formula to convert a raw score into a standardized score. We need to find the Z-score that corresponds to the 83.4th percentile, as 16.6 percent is the area to the left of this score. We can then use the Z-score formula to find the corresponding raw score.
Z = (X - μ) / σ
Where: Z is the Z-score, X is the raw score, μ is the mean, and σ is the standard deviation. Rearranging the formula, we have:
X = (Z * σ) + μ
Since the mean is 78 and the standard deviation is 10, we substitute the values into the formula:
X = (Z * 10) + 78
Next, we need to find the Z-score that corresponds to the 83.4th percentile using a Z-score table or a calculator. From the table, we find that the Z-score is approximately 0.9998. Substituting this value into the formula, we can solve for X:
X = (0.9998 * 10) + 78
X = 9.998 + 78
X ≈ 87.998
Therefore, the minimum score needed to receive a grade of A is approximately 88.
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Can anyone help me please ASAP
Given:
It is given that the measurements of the triangle.
The measure of ∠2 is (3x + 3)°
The measure of ∠3 is (3x - 4)°
The measure of ∠4 is (5x + 8)°
We need to determine the measure of ∠1 and ∠4.
Value of x:
The value of x can be determined using the exterior angle theorem.
Applying the theorem, we have;
[tex]m \angle 4=m \angle 2+m \angle 3[/tex]
Substituting the values, we get;
[tex]5x+8=3x+3+3x-4[/tex]
[tex]5x+8=6x-1[/tex]
[tex]-x+8=-1[/tex]
[tex]-x=-9[/tex]
[tex]x=9[/tex]
Thus, the value of x is 9.
Measure of ∠4:
Substituting the value of x in the expression of ∠4, we get;
[tex]m\angle 4=5(9)+8[/tex]
[tex]=45+8[/tex]
[tex]m\angle 4=53^{\circ}[/tex]
Thus, the measure of ∠4 is 53°
Measure of ∠1:
The angles 1 and 4 are linear pairs and hence these angles add up to 180°
Thus, we have;
[tex]\angle 1+ \angle 4=180^{\circ}[/tex]
Substituting the values, we get;
[tex]\angle 1+ 53^{\circ}=180^{\circ}[/tex]
[tex]\angle 1=127^{\circ}[/tex]
Thus, the measure of ∠1 is 127°
8. Lara subtracts 73 from 188. Which one of these
steps should she follow?
o Ungroup 8 tens as 7 tens 10 ones.
o Subtract 3 ones from 8 tens.
o Subtract 7 tens from 8 tens.
Answer: o Subtract 7 tens from 8 tens.
Step-by-step explanation: you subtract
Answer:
Step-by-step explanation:subtract 7 tens from 8 tens
Which prism has an area of 6 cubic units?
A prism has a length of 1 and one-half, height of 1, and width of 3.
A prism has a length of 2, height of 1 and one-half, and width of 2.
A prism has a length of 1 and one-fourth, height of 1, and width of 4.
A prism has a length of 3, height of 1 and one-fourth, and width of 2.
Answer:
A prism that has a length of 2, height of 1 and one-half, and width of 2.
Step-by-step explanation:
The formula for calculating the area of a prism is base area × height
Volume = L×W×H
L is the length of the prism
W is the width
H is the height.
To determine the prism that has an area of 6 cubic units, we will substitute the values in the option in the formula.
Using option D i.e prism that has a length of 2, height of 1 and one-half, and width of 2.
L = 2units, H = 1 1/2 units , W = 2units
Substituting in the formula for finding the volume
V = 2×2×3/2
Volume of the prism = 6cubic units
Answer:
B
Step-by-step explanation:
In a random sample of n1 = 156 male Statistics students, there are x1 = 81 underclassmen. In a random sample of n2 = 320 female Statistics students, there are x2 = 221 underclassmen. The researcher would like to test the hypothesis that the percent of males who are underclassmen stats students is less than the percent of females who are underclassmen stats students. What is the p-value for the test of hypothesis? i.e. Find P(Z < test statistic). Enter your answer to 4 decimal places.
Answer:
The p-value for the test of hypothesis is P(z<-3.617)=0.0002.
Step-by-step explanation:
Hypothesis test on the difference between proportions.
The claim is that the percent of males who are underclassmen stats students (π1) is less than the percent of females who are underclassmen stats students (π2).
Then, the null and alternative hypothesis are:
[tex]H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2<0[/tex]
The male sample has a size n1=156. The sample proportion is p1=81/156=0.52.
The female sample has a size n2=221. The sample proportion in this case is p2=221/320=0.69.
The weigthed average of proportions p, needed to calculate the standard error, is:
[tex]p=\dfrac{n_1p_1+n_2p_2}{n_1+n_2}=\dfrac{81+221}{156+320}=\dfrac{302}{476}= 0.63[/tex]
The standard error for the difference in proportions is:
[tex]\sigma_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.63*0.37}{156}+\dfrac{0.63*0.37}{320}}\\\\\\\sigma_{p1-p2}=\sqrt{\dfrac{0.2331}{156}+\dfrac{0.2331}{320}}=\sqrt{0.001503871+0.000728438}=\sqrt{0.002232308}\\\\\\\sigma_{p1-p2}=0.047[/tex]
Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p_1-p_2}{\sigma_{p1-p2}}=\dfrac{0.52-0.69}{0.047}=\dfrac{-0.17}{0.047}=-3.617[/tex]
The P-value for this left tailed test is:
[tex]P-value = P(z<-3.617)=0.00015[/tex]
Answer:
[tex]z=\frac{0.519-0.691}{\sqrt{0.634(1-0.634)(\frac{1}{156}+\frac{1}{320})}}=-3.657[/tex]
[tex]p_v =P(Z<-3.657)=0.0001[/tex]
Step-by-step explanation:
Data given and notation
[tex]X_{1}=81[/tex] represent the number of males underclassmen
[tex]X_{2}=221[/tex] represent the number of females underclassmen
[tex]n_{1}=156[/tex] sample of male
[tex]n_{2}=320[/tex] sample of female
[tex]p_{1}=\frac{81}{156}=0.519[/tex] represent the proportion of males underclassmen
[tex]p_{2}=\frac{221}{320}= 0.691[/tex] represent the proportion of females underclassmen
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the value for the test (variable of interest)
Concepts and formulas to use
We need to conduct a hypothesis in order to check if the percent of males who are underclassmen stats students is less than the percent of females who are underclassmen stats students , the system of hypothesis would be:
Null hypothesis:[tex]p_{1} \geq p_{2}[/tex]
Alternative hypothesis:[tex]p_{1} < p_{2}[/tex]
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex] (1)
Where [tex]\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{81+221}{156+320}=0.634[/tex]
Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.519-0.691}{\sqrt{0.634(1-0.634)(\frac{1}{156}+\frac{1}{320})}}=-3.657[/tex]
Statistical decision
For this case we don't have a significance level provided [tex]\alpha[/tex], but we can calculate the p value for this test.
Since is a one side test the p value would be:
[tex]p_v =P(Z<-3.657)=0.0001[/tex]
Traders often buy foreign currency in hope of making money when the currency's value changes. For example, on a particular day, one U.S. dollar could purchase 0.8869 Euros, and one Euro could purchase 143.1126 yen. Let fix) represent the number of Euros you can buy with x dollars, and let g(x) represent the number of yen you can buy with x Euros. (a) Find a function that relates dollars to Euros fx)Simplify your answer.) (b) Find a function that relates Euros to yen. gxSimplify your answer.) (c) Use the results of parts (a) and (b) to find a function that relates dollars to yen. That is, find (g o f)(x)-g(fx g(f(x)) Simplify your answer. Use integers or decimals for any numbers in the expression. Round to four decimal places as needed.) (d) What is g(1000))? g(f(1000)) Type an integer or decimal rounded to one decimal place as needed.)
Answer:
(a)f(x)=0.8869x
(b)g(x)=143.1126x
(c)g(f(x))=126.9266x
(d)g(f(1000))=126926.6 Yen
Step-by-step explanation:
Given on a particular day
One U.S. dollar could purchase 0.8869 EurosOne Euro could purchase 143.1126 yen(a)If x represents the number of Dollars
Since one can purchase 0.8869 Euro with 1 USD, the function f(x) is a direct relationship where x is dollars and f(x) is in Euros.
f(x)=0.8869x(b)If x represents the number of Euros
Since one can purchase 143.1126 yen with 1 Euros, the function g(x) is a direct relationship where x is Euros and g(x) is in Yen.
g(x)=143.1126x(c)Given:
g(x)=143.1126xf(x)=0.8869xg(f(x))=143.1126(0.8869x)
g(f(x))=126.9266x(d)g(f(1000))
g(f(x))=126.9266xg(f(1000))=126.9266 X 1000 =126926.6 YenBrooklyn bought 1 pound of cucumbers for a salad. She bought twice as much lettuce. How many ounces of lettuce did Brooklyn buy for the salad.
Answer:
32 ounces
Step-by-step explanation:
She bought 1 cucumber. She bought twice as much lettuce.
1(2) = 2 lbs of lettuce.
There are 16 ounces to the lb.
2 (16 ounces) = 32 ounces
Answer:
She bought 32 oz of lettuce.
Step-by-step explanation:
There are 16 oz in 1 lb. twice as much means 2x. 2 x 16 = 32.
Please help me in don't understand how to do this
Answer:
36
Step-by-step explanation:
[tex] \frac{c}{4} - 5 = 4 \\ \\ \frac{c}{4} = 4 + 5\\ \\ \frac{c}{4} = 9 \\ \\ c = 9 \times 4 \\ \\ \huge \red{ \boxed{ c = 36}}[/tex]
Which of the following are examples of limiting factors?
food, water, cell composition
cell composition water sunlight
food water sunlight
sizer water sunlight
simplified expression -6x+2/3(9-15x)-2
Answer:
-16x+4
Step-by-step explanation:
-6x+2/3(9-15x)-2
Distribute
-6x +2/3 *9 +2/3*(-15x) -2
-6x +6 -10x -2
Combine like terms
-6x-10x +6-2
-16x+4
You are at a campus party where there are a total number of n people. The host asked everyone to put their phones in a bowl while walking in. A noise complaint ends the party abruptly, and everyone heads for the door, hastily grabbing their phones from the bowl Assume every guest has one and exactly one phone, and that they pick a phone at random (so that every assignment of a phone to a person is equally likely). What is the probability that: a. Every person gets their phone back? b. The first m persons to pick each get their own phones back? c. The first m persons to pick each get a phone belonging to the last m persons to pick? Hint: Try this thought experiment with a few choices of mand n to get a feel for the numbers that show up.)
Answer:
1, [tex]\frac{m}{n}[/tex], [tex]\frac{1-m}{n}[/tex].
Step-by-step explanation:
probability = [tex]\frac{Number of Possible Outcomes}{Total Outcomes}[/tex]
Total number of persons in the party = n
a) Pr ( every person gets their phone back) = Pr (each person picks his phone ) multiplied by number of person
= [tex]\frac{1}{n}[/tex] × n = 1.
No of first m persons to pick = m
No of last m persons to pick = 1 - m
b) Pr (first m persons to pick each gets their phones back) = [tex]\frac{m}{n}[/tex]
c) Pr( first m persons get a phone belonging to last m persons) = [tex]\frac{1-m}{n}[/tex]
A student takes a true-false test that has 10 questions and guesses randomly at each answer. Let X be the number of questions answered correctly. Find P(Fewer than 3). Round your answer to 2 decimal places.
Answer:
P(Fewer than 3) = 0.05.
Step-by-step explanation:
We are given that a student takes a true-false test that has 10 questions and guesses randomly at each answer.
The above situation can be represented through Binomial distribution;
[tex]P(X=r) = \binom{n}{r}p^{r} (1-p)^{n-r} ; x = 0,1,2,3,.....[/tex]
where, n = number of trials (samples) taken = 10 questions
r = number of success = fewer than 3
p = probability of success which in our question is probability
that question is answered correctly, i.e; 50%
LET X = Number of questions answered correctly
So, it means X ~ Binom(n = 10, p = 0.50)
Now, Probability that Fewer than 3 questions are answered correctly is given by = P(X < 3)
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
= [tex]\binom{10}{0}\times 0.50^{0} \times (1-0.50)^{10-0}+ \binom{10}{1}\times 0.50^{1} \times (1-0.50)^{10-1}+ \binom{10}{2}\times 0.50^{2} \times (1-0.50)^{10-2}[/tex]
= [tex]1 \times 0.50^{10} + 10 \times 0.50^{10} +45 \times 0.50^{10}[/tex]
= 0.05
Hence, the P(Fewer than 3) is 0.05.
To find the probability of the student passing the test with at least a 70 percent, we can use the binomial probability formula. The probability of the student passing the test with at least 70 percent is 0.1719 (rounded to 2 decimal places).
Explanation:To find the probability of the student passing the test with at least a 70 percent, we need to find the probability of the student answering 7, 8, 9, or 10 questions correctly out of the 10 questions. Since the student randomly guesses at each answer, the probability of guessing correctly is 0.5. Now we can calculate the probability using the binomial probability formula:
P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
P(X = k) = C(10, k) * (0.5)^k * (0.5)^(10-k), where C(n, r) is the binomial coefficient (n choose r).
Calculating each probability and summing them up, we get P(X ≥ 7) = 0.171875. Therefore, the probability of the student passing the test with at least 70 percent is 0.1719 (rounded to 2 decimal places).
Is 8 a solution to 3x + 9 = 13?
Answer:
No
Step-by-step explanation:
3x + 9 = 13
Subtract 9 from each side
3x + 9-9 = 13-9
3x = 4
Divide each side by 3
3x/3 = 4/3
x = 4/3
8 is not a solution
A study of 178 cases of disease X were identified from a state registry. A total of 220 control subjects were then recruited from random-digit dial procedure. 16 cases had been exposed, compared to only 8 controls. How likely were cases to report an exposure compared with controls
Answer:
2.47 times more likely.
Step-by-step explanation:
16 out of 178 cases were reported for exposure.
And 8 out of 220 control reported for exposure.
Chances that a case would be reported for exposure = (16/178) = 0.0898876404
Chances that one control would report for exposure = (8/220) = 0.0363636364
Comparing both, how likely were cases to report an exposure compared with controls
= (0.0898876404) ÷ (0.0363636364)
= 2.4719101085 = 2.47 times more likely.
Hope this Helps!!
Decompose fraction 2 3/4
To decompose the fraction 2 3/4, convert it to an improper fraction by multiplying the whole number by the denominator of the fraction, add the numerator, and place over original denominator, resulting in 11/4.
Explanation:The question asks to decompose the fraction 2 3/4 into its components. To decompose this mixed number, we need to convert it to an improper fraction. The process involves multiplying the whole number by the denominator of the fraction part, adding the numerator of the fraction part, and then placing the result over the original denominator.
Therefore, the mixed number 2 3/4 decomposed into an improper fraction is 11/4.
Evaluate 6.5b - 12.03 when b= 3
Answer:
7.47
Step-by-step explanation:
6.5b - 12.03
Let b=3
6.5(3) - 12.03
Multiply first
19.5 - 12.03
7.47
Hi I think your answer is - 5.5
6.5b b=3. So you replace B with 3 and that makes it 6.53-12.03.
wich gives you - 5.5.
Find the x-coordinates of all critical points of the given function. Determine whether each critical point is a relative maximum, minimum, or neither by first applying the second derivative test, and, if the test fails, by some other method. f(x) = 5x^6 − 10x^4
Answer:
x = 0, local maximumx = ±(2/3)√3, global minimaStep-by-step explanation:
The first derivative is ...
f'(x) = 30x^5 -40x^3 = 10x^3(3x^2 -4)
This will have zeros (critical points) at x=0 and x=±√(4/3).*
We don't need the second derivative to tell the nature of these critical points. Since the degree is even, the function is symmetrical about x=0. Since the leading coefficient is positive, it generally has a U-shape. This means the "outer" critical points will be minima, and the central one will be a local maximum.
__
However, since we're asked to use the 2nd derivative test first, we find the 2nd derivative to be ...
f''(x) = 150x^4 -120x^2 = 30x^2(5x^2 -4)
For x=0, f''(0) = 0 -- as we expect for a function with a high multiplicity of the root at that point. For x either side of zero, both the function and the second derivative are negative, indicating downward concavity. That is, x = 0 is a local maximum.
For x² = 4/3, the second derivative is positive, indicating upward concavity. At x = ±√(4/3), we have local minima.
_____
* The "simplified" equivalent to √(4/3) is (2/3)√3.
The critical points of the function f(x) = 5x^6 − 10x^4 are x = 0, x = ±√(4/3). The point at x = 0 is a relative maximum while the points at x = ±√(4/3) are relative minima based on the second derivative test.
Explanation:Given the function f(x) = 5x^6 − 10x^4, we first find the critical points. This is done by finding the derivative of the function and setting it equal to zero. For this function, the derivative, f'(x), is 30x^5 - 40x^3 = 0. Solving this equation for x, we get x = 0, and x = ±√(4/3).
Next, we apply the second derivative test by taking the second derivative of the original function, f''(x). This gives us f''(x) = 150x^4 - 120x^2. We substitute the obtained critical points into the second derivative. If f''(x) > 0, then the point is a relative minimum, if f''(x) < 0, it's a relative maximum. If neither, we need to consider higher order derivatives or other methods.
The second derivative is negative at x = 0, so that position is a relative maximum. The second derivative is positive at x = ±√(4/3), so these positions are relative minima.
Learn more about Critical Points here:https://brainly.com/question/32077588
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Hotel cost 60 per night flight cost 150 has a budget of 500 how many nights can she afford
Answer:
3 nights
Step-by-step explanation:
because 1 flight there and one flight back =300 then add 3 nights =480
Answer:
5 nights or less
Step-by-step explanation:
You can do this by writing an inequality and solving it.
Let n = number of nights.
1 hotel night costs $60. n number of hotel nights cost 60n.
The flight costs $150.
The total cost is the price of the hotel plus the price of the flight.
60n + 150
The total price must be less than or equal to $500.
[tex] 60n + 150 \le 500 [/tex]
Now we solve the inequality.
Subtract 150 from both sides.
[tex] 60n \le 350 [/tex]
Divide both sides by 60.
[tex] n \le \dfrac{350}{60} [/tex]
350 divided by 60 is 5.8333...
[tex] n \le 5.8 [/tex]
The number of night is less than or equal to 5.8, and it must be a whole number, so the most number of nights she can afford is 5.
7,945\100 Is the equal as which number?
Answer:
79.45
Step-by-step explanation:
An aerosol can contains gases under a pressure
of 4.5 atm at 24 ◦C. If the can is left on a
hot sandy beach, the pressure of the gases
increases to 4.66 atm. What is the Celsius
temperature on the beach?
Answer:
temperature on the beach = T2 = 34.56 °C
Step-by-step explanation:
We are given;
P1 = 4.5 atm
T1 = 24 °C = 24 + 273 = 297 K
P2 = 4.66 atm
Thus, P1/T1 = P2 /T2
So, T2 = P2•T1/P1
Thus, T2 = (4.66x 297)/4.5
T2 = 307.56 K
Let's convert to °C to obtain ;
T2 = 307.56 - 273
T2 = 34.56 °C