(a) 1000
(b) 11100
(c) 11095.
Step-by-step explanation:
(a) If A1 is a subset of A2 and A2 is a subset of A3, then all the elements of A1 are in A2 and all the elements of A2 are in A3.
Then, n(A1 n A2) = 100, n(A2 n A3) = 1000 , n(A1 n A3) = 100 and n(A1 n A2 n A3) = 100.
So, we get
[tex]n(A1\cup A2\cup A3)\\\\=n(A1)+n(A2)+n(A3)-n(A1\cap A2)-n(A2\cap A3)-n(A1\cap A3)+n(A1\cap A2\cap A3)\\\\=100+1000+1000-100-1000-100+100\\\\=1000.[/tex]
(b) If the sets are pairwise disjoint, then
n(A1 n A2) = n(A2 n A3) = n(A1 n A3) = n(A1 n A2 n A3) = 0.
So, we get
[tex]n(A1\cup A2\cup A3)\\\\=n(A1)+n(A2)+n(A3)\\\\=100+1000+10000\\\\=11100.[/tex]
(c) If there are two elements common to each pair of sets and one element in all three sets, then
n(A1 n A2) = 2, n(A2 n A3) = 2, n(A1 n A3) = 2 and n(A1 n A2 n A3) = 1.
So, we get
[tex]n(A1\cup A2\cup A3)\\\\=n(A1)+n(A2)+n(A3)-n(A1\cap A2)-n(A2\cap A3)-n(A1\cap A3)-n(A1\cap A2\cap A3)\\\\=100+1000+1000-2-2-2+1\\\\=11100-5\\\\=11095.[/tex]
Final answer:
The number of elements in the union of sets A1, A2, and A3 varies depending on their relationships. For subsets (a), the count is 10,000; for disjoint sets (b), it is 11,100; and when each pair has common elements plus one common to all (c), the count is 11,095.
Explanation:
Finding the Number of Elements in the Union of Sets
To find the number of elements in the union of sets A1, A2, and A3, we need to consider the given conditions.
a) A1 ⊆ A2 and A2 ⊆ A3
Since A1 is a subset of A2, and A2 is a subset of A3, all elements of A1 and A2 are included in A3. Therefore, the
number of elements in A1 ∪ A2 ∪ A3 equals the number of elements in A3, which is 10,000.
b) The Sets Are Pairwise Disjoint
If the sets are pairwise disjoint, this means they share no elements in common. We simply add the number of elements in each set to find the union's total count. This gives us 100 + 1000 + 10,000 = 11,100 elements in the union.
c) Two Elements Common to Each Pair and One in All Three
With two elements common to each pair of sets and one element in all three, we need to subtract the common elements to avoid double-counting. So, A1 ∪ A2 ∪ A3 will have 100 + 1000 + 10,000 - 2 - 2 - 2 + 1 (since 1 element is counted three times, we add it back once) which equals 11,095 elements.
A large explosion causes wood and metal debris to rise vertically into the air with an initial velocity of 160 feet per second. The function h(t) = 160 t − 16 t 2 160t-16t2 gives the height of the falling debris above the ground, in feet, t t seconds after the explosion.
a. Use the given polynomial to find the height of the debris 2 second(s) after the explosion.
b. Factor the given polynomial completely.
Answer:
a) The debris was 256 feet into the air after 2 seconds of the explosion.
b)
[tex]h(t) = -16t(t-10)[/tex]
Step-by-step explanation:
We are given the following in the question:
Initial Velocity = 160 feet per second
[tex]h(t) = 160t-16t^2[/tex]
The above function gives the height in feet and t is seconds after the explosion.
a) Height of the debris 2 second(s) after the explosion.
We put t = 2 in the above function
[tex]h(2) = 160(2)-16(2)^2 = 256[/tex]
Thus, the debris was 256 feet into the air after 2 seconds of the explosion.
b) Factor the polynomial
[tex]h(t) = 160t-16t^2\\= 16t(10-t)\\=-16t(t-10)[/tex]
Final answer:
The height of the debris 2 seconds after the explosion is 256 feet. The given polynomial can be factored completely as -16t(t - 10)
Explanation:
To find the height of the debris 2 seconds after the explosion, we can substitute t = 2 into the equation h(t) = 160t - 16t^2.
So, h(2) = 160(2) - 16(2)^2 = 320 - 16(4) = 320 - 64 = 256 feet.
Therefore, the height of the debris 2 seconds after the explosion is 256 feet.
To factor the given polynomial completely, we can rewrite it as:
h(t) = -16t^2 + 160t.
Now, we can factor out a common factor of -16t:
h(t) = -16t(t - 10).
This gives us the completely factored form of the polynomial.
In the game of Dubblefud, red chips, blue chips and green chips are each worth 2, 4 and 5 points respectively. In a certain selection of chips, the product of the point values of the chips is 16,000. If the number of blue chips in this selection equals the number of green chips, how many red chips are in the selection?A. 1
B. 2
C. 3
D. 4
E. 5
Final answer:
The problem is solved by finding the prime factorization of 16,000 and assigning the factors to the chips based on their point values, resulting in there being 3 red chips.
Explanation:
The student's question is a typical problem in combinatorics, a field of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. This problem also involves prime factorization as a method to solve for the number of chips.
To solve the problem, let us denote by R, B, and G the number of red, blue, and green chips respectively. Since the blue chips and green chips have the same quantity, the problem can be solved by first finding the prime factorization of the point value product of 16,000, which is 2⁷ × 5³, and then distributing these prime factors to match the point values of the chips.
Since the point values of blue and green chips are 4 (2²) and 5 respectively and their quantities are equal, we match the prime factors of 5 first. There are 3 factors of 5, so we assign one to each blue and green chip, resulting in 1 remaining.
Then, we match 2² or 4 to each blue and green chip, using 4 out of 7 factors of 2, leaving us with 3 factors of 2, which can be matched with 3 red chips since they have a value of 2 each.
Therefore, we have 3 red chips which is answer option C.
Jacob distributed a survey to his fellow students asking them how many hours they'd spent playing sports in the past day. He also asked them to rate their mood on a scale from 000 to 101010, with 101010 being the happiest. A line was fit to the data to model the relationship.
Jacob's survey is a study in statistics, specifically looking at the correlation between the amount of time spent on sports and student's mood ratings. A line is fit to the data to determine the relationship, with the direction of the line offering insights into how these two variables correlate, but this does not imply causation.
Explanation:From your question, Jacob performed a survey asking about the number of hours students spent playing sports in the past day and asked them to rate their mood. It's a study of basic statistics, specifically focusing on correlation between two variables, here those are the number of hours spent on sports and mood ratings. To determine a relationship between these variables, a line is often fit to the data, using methods like linear regression.
For example, if the line on the graph is rising, it indicates a positive correlation between the amount of sports played and a student's mood, meaning that as sports playtime goes up, so does mood ratings. A falling line means there's a negative correlation. If there's no clear direction, it's likely that there's no significant correlation between the two variables. But remember that correlation doesn't mean causation: just because two things correlate doesn't mean that one causes the other.
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Each person in a simple random sample of 1,800 received a survey, and 267 people returned their survey. How could nonresponse cause the results of the survey to be biased?
-Those who did not respond reduced the sample size, and small samples have more bias than large samples.
- Those who did not respond caused a violation of the assumption of independence.
-Those who did not respond are indistinguishable from those who did not receive the survey.
-Those who did not respond may differ in some important way from those who did respond.
-Those who did not respond represent a stratum, changing the random sample into a stratified random sample.
Nonresponse can cause bias in survey results due to differences in characteristics, the representation of strata, and the smaller sample size.
Explanation:Nonresponse can cause the results of a survey to be biased in several ways:
Those who did not respond may differ in some important way from those who did respond. This means that the characteristics of the nonresponders may be different from the characteristics of the responders, leading to biased results.Those who did not respond may represent a stratum, changing the random sample into a stratified random sample. This can introduce bias if the nonresponders differ from the rest of the population in a systematic way.Those who did not respond may reduce the sample size, and small samples have more bias than large samples. When the sample size is small, the results may not accurately reflect the population.Overall, nonresponse can introduce bias into a survey by excluding certain individuals or groups from the sample, leading to potentially inaccurate and biased results.
Han spent 75 minutes practicing the piano over the weekend. Priya practiced the violin for 152% as much as Han practiced the piano. How long did she practice
Priya practiced the violin for 114 minutes
Solution:Given that
Han spent 75 minutes practicing the piano over the weekend
Priya practiced the violin for 152% as much as Han practiced the piano
Need to determine how long did priya practice
Duration of Han practicing the piano over weekend = 75 minutes
As given that Priya practiced the violin for 152% as much as Han practiced the piano
=> Duration of Priya practicing the piano = 152% of Han practicing the piano
=> Duration of Priya practicing the piano = 152% of 75 minutes
We know that a % of b is written in fraction as [tex]\frac{a}{100} \times b[/tex]
[tex]\Rightarrow \text { Duration of Priya practicing the piano }=\frac{152}{100} \times 75=114 \text { minutes }[/tex]
Hence Priya practiced the violin for 114 minutes
The 8 leaders of the G8 nations convene in Rome and stand in a row as they get ready to have some pictures of them taken by the press. What is the probability that the picture that the New York Times' editors will randomly select for publishing the next day is one in which Berlusconi is not standing next to Obama? (assuming that there are pictures of all possible standing arrangements).
Answer:
The required probability is [tex]\frac{3}{4}[/tex].
Step-by-step explanation:
Consider the provided information.
There are 8 leaders.
Thus, the total number of ways to arrange 8 leaders are 8!.
Assume that Obama and Berlusconi is one person.
Therefore the total number of leaders are 7 (As Obama and Berlusconi is one person).
The number of ways in which 7 leader can be arranged: 7!
Although Obama and Berlusconi is one unit but they can interchange their place in 2 ways. Like Obama and Berlusconi or Berlusconi and Obama
That means the total number of ways : 7!×2
The number of ways in which they are not next to each other = Total number of ways - The number of ways in which they are next to each other
Number of ways they are not next to each other = 8!-7!×2
The probability that they are not next to each other = [tex]\frac{8!-7!\times2}{8!}=\frac{3}{4}[/tex]
Hence, the required probability is [tex]\frac{3}{4}[/tex].
Solve for x (log equation) (don’t mind the work)
Answer:
Step-by-step explanation:
University officials say that at least 70% of the voting student population supports a fee increase. If the 95% confidence interval estimating the proportion of students supporting the fee increase is [0.75, 0.85], what conclusion can be drawn?
A. 70% is not in the interval, so another sample is needed.
B.70% is not in the interval, so assume it will not be supported.
C.The interval estimate is above 70%, so infer that it will be supported.
D.Since this was not based on population, we cannot make a conclusion.
the correct answer is C. The interval estimate is above 70%, so infer that it will be supported.The 95% confidence interval (0.75, 0.85) indicates that we can expect around 75% to 85% of students to support the fee increase, which is more than the presumed 70%. So, we can infer that the fee increase will be supported.
Explanation:In the scenario given, the 95% confidence interval for the proportion of students supporting the fee increase is [0.75, 0.85].
This means that we are 95% confident that the true proportion of students who support the fee increase falls within this interval. Since the entire range is above 70%, we can be pretty confident that more than 70% of the student population supports the fee increase.
The confidence interval is a statistical method that gives us a range in which the true parameter likely falls based on our sample data. This shows us that statistical inference can give us insights about a population based on a sample.
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a. Find PR in the diagram
b. Find the perimeter of quadrilateral PQRS
Answer:
Step-by-step explanation:
The quadrilateral has 4 sides and only two of them are equal.
A) to find PR, we will consider the triangle, PRQ.
Using cosine rule
a^2 = b^2 + c^2 - 2abcos A
We are looking for PR
PR^2 = 8^2 + 7^2 - 2 ×8 × 7Cos70
PR^2 = 64 + 49 - 112 × 0.3420
PR^2 = 113 - 38.304 = 74.696
PR = √74.696 = 8.64
B) to find the perimeter of PQRS, we will consider the triangle, RSP. It is an isosceles triangle. Therefore, two sides and two base angles are equal. To determine the length of SP,
We will use the sine rule because only one side,PR is known
For sine rule,
a/sinA = b/sinB
SP/ sin 35 = 8.64/sin110
Cross multiplying
SPsin110 = 8.64sin35
SP = 8.64sin35/sin110
SP = (8.64 × 0.5736)/0.9397
SP = 5.27
SR = SP = 5.27
The perimeter of the quadrilateral PQRS is the sum of the sides. The perimeter = 8 + 7 + 5.27 + 5.27 = 25.54 cm
Simplify the rational expressions. state any excluded values.
1. 2x-8/x-4
2. 4x-8/4x+20
3. x+7/x^2+4x-21
4. x^2-3x-10/x+2
5. x^2-4/2-x
I need help pleeeese
See the answers in explanation
Explanation:Let's solve this problem as follows:
First.[tex]\bullet \ \frac{2x-8}{x-4} \\ \\ Common \ factor \ 2 \ from \ the \ numerator: \\ \\ \frac{2x-8}{x-4}=\frac{2(x-4)}{x-4} =2[/tex]
Second.[tex]\bullet \ \frac{4x-8}{4x+20} \\ \\ Common \ factor \ 4 \ from \ the \ numerator \ and \ denominator: \\ \\ \frac{4x-8}{4x+20}=\frac{4(x-2)}{4(x+5)}=\frac{(x-2)}{(x+5)}[/tex]
Third[tex]\bullet \ \frac{x+7}{x^2+4x-21} \\ \\ Rearranging \ denominator: \\ \\ \frac{x+7}{x^2-3x+7x-21}=\frac{x+7}{x(x-3)+7(x-3)}=\frac{x+7}{x(x-3)+7(x-3)} \\ \\ Common \ factor \ x-3 \ from \ denominator: \\ \\ \frac{x+7}{(x-3)(x+7)}=\frac{1}{x-3}[/tex]
Fourth.[tex]\bullet \ \frac{x^2-3x-10}{x+2} \\ \\ Rearranging \ numerator: \\ \\ \frac{x^2-3x-10}{x+2}=\frac{x^2-5x+2-10}{x+2}=\frac{x(x-5)+2(x-5)}{x+2} \\ \\ Common \ factor \ x-5 \\ \\ \frac{(x-5)(x+2)}{x+2}=x-5[/tex]
Fifth.[tex]\bullet \ \frac{x^2-4}{2-x} \\ \\ Difference \ of \ squares \ from \ numerator: \\ \\ \frac{(x-2)(x+2)}{2-x} \\ \\ Common \ factor \ -1 \ from \ denominator: \\ \\ \frac{(x-2)(x+2)}{-(x-2)}=-(x+2)[/tex]
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In a recent Super Bowl, a TV network predicted that 50 % of the audience would express an interest in seeing one of its forthcoming television shows. The network ran commercials for these shows during the Super Bowl. The day after the Super Bowl, and Advertising Group sampled 106 people who saw the commercials and found that 48 of them said they would watch one of the television shows.Suppose you are have the following null and alternative hypotheses for a test you are running:H0:p=0.5Ha:p≠0.5Calculate the test statistic, rounded to 3 decimal placesz=
Answer:
z= -0.968
We can conclude that we fail to reject the null hypothesis, and we can said that at 5% of significance the proportion of people who says that they would watch one of the television shows not differs from 0.5 or 50% .
Step-by-step explanation:
1) Data given and notation n
n=106 represent the random sample taken
X=48 represent the people who says that they would watch one of the television shows.
[tex]\hat p=\frac{48}{106}=0.453[/tex] estimated proportion of people who says that they would watch one of the television shows.
[tex]p_o=0.5[/tex] is the value that we want to test
[tex]\alpha[/tex] represent the significance level
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that 50% of people who says that they would watch one of the television shows.:
Null hypothesis:[tex]p=0.5[/tex]
Alternative hypothesis:[tex]p \neq 0.5[/tex]
When we conduct a proportion test we need to use the z statisitc, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
3) Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.453 -0.5}{\sqrt{\frac{0.5(1-0.5)}{106}}}=-0.968[/tex]
4) Statistical decision
P value method or p value approach . "This method consists on determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The significance level is not provided, but we can assume [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.
Since is a bilateral test the p value would be:
[tex]p_v =2*P(z<-0.968)=0.333[/tex]
So based on the p value obtained and using the significance level assumed [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we fail to reject the null hypothesis, and we can said that at 5% of significance the proportion of people who says that they would watch one of the television shows not differs from 0.5 or 50% .
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Two wires help support a pole. The wire at point A forms an angle of 54° with the ground and the wire at point B forms an angle of 72° with the ground. The distance between the wires on the ground is 23 m. Find the height of the pole to the nearest tenth of a meter.
Height of the pole (DC) is 57.2709m
Step-by-step explanation:
Here, Wire DA and Wire DB supports a pole.
Given that Angle, A=54 , B= 72.
Also, AB = 23m
Now, Taking triangle BCD and Using basic trigonometry
Height of pole H = DC
[tex]TanB = \frac{DC}{BC}[/tex]
[tex]BC= \frac{DC}{TanB}[/tex]
Now, Taking triangle ACD and Using basic trigonometry
[tex]TanA = \frac{DC}{AC}[/tex]
[tex]AC= \frac{DC}{TanA}[/tex]
From figure, we know that
AC = AB + BC
AC - BC = AB = 23
Replacing values of AC and BC
[tex]\frac{DC}{TanA} - \frac{DC}{TanB}=23\\
DC(\frac{1}{TanA}-\frac{1}{TanB})=23[/tex]
Now, TanB= Tan72 =3.0776 and TanA = Tan54=1.3763
[tex]DC (\frac{1}{1.3763} - \frac{1}{3.0776})_= 23[/tex]
[tex]DC ( 0.7265-0.3249)= 23[/tex]
[tex]DC ( 0.4016 )= 23[/tex]
[tex]DC = 57.2709 [/tex]
Thus, Height of thepole is 57.2709m
The hypotenuse of a right triangle has one end at the origin and one end on the curve y = x 2 e −3x , with x ≥ 0. One of the other two sides is on the x-axis, the other side is parallel to the y-axis. Find the maximum area of such a triangle. At what x-value does it occur?
Answer:
At x = 1 and maximum area = 0.0499
Step-by-step explanation:
The hypotenuse of a right triangle has one end at the origin and other end on the curve, [tex]y=x^2e^{-3x}[/tex] with x ≥ 0.
One leg of right triangle is x-axis and another leg parallel to y-axis.
Length of base of right triangle = x
Height of right triangle = y
Area of right triangle, [tex]A=\dfrac{1}{2}xy[/tex]
[tex]A=\dfrac{1}{2}x^3e^{-3x}[/tex]
For maximum/minimum value of area.
[tex]\dfrac{dA}{dx}=\dfrac{3}{2}x^2e^{-3x}-\dfrac{3}{2}x^3e^{-3x}[/tex]
Now, find critical point, [tex]\dfrac{dA}{dx}=0[/tex]
[tex]\dfrac{3}{2}x^2e^{-3x}-\dfrac{3}{2}x^3e^{-3x}=0[/tex]
[tex]\dfrac{3}{2}x^2e^{-3x}(1-x)=0[/tex]
x =0,1
For x = 0, y = 0
For x = 1, [tex]y=e^{-3}[/tex]
using double derivative test:-
[tex]\dfrac{d^2A}{dx^2}=\dfrac{6}{2}xe^{-3x}-\dfrac{9}{2}x^2e^{-3x}-\dfrac{9}{2}x^2e^{-3x}-\dfrac{9}{2}x^3e^{-3x}[/tex]
At x= 0 , [tex]\dfrac{d^2A}{dx^2}=0[/tex]
Neither maximum nor minimum
At x = 1, [tex]\dfrac{d^2A}{dx^2}=-0.14<0[/tex]
Maximum area at x = 1
The maximum area of right triangle at x = 1
Maximum area, [tex]A=\dfrac{1}{e^3}\approx 0.0499[/tex]
The point of maxima will be x=3 and the maximum area will be 0.002 square units.
According to the diagram attached
The area of the given triangle will be = 0.5*base*height
As one end of the hypotenuse is on the curve [tex]y = x^2e^(-3x)[/tex], Coordinates of one end of the hypotenuse will be [tex](x, x^2e^(-3x)[/tex].
Area A(x) of the given triangle = 0.5*base* height
Base = x
Height = [tex]x^2e^(-3x)[/tex]
So A(x) = [tex]0.5*x*x^{2} *e^(-3x)[/tex]
[tex]A(x) = 0.5*x*x^{2} *e^(-3x)\\\\A(x) = 0.5 x^3e^(-3x)[/tex]
For the maximum area,
[tex]A'(x) = 0\\\\x^2e^(-3x) (x-3) = 0\\x = 0 and x=3[/tex]will be the points of extremum.
What are the points of the extremum?
Points of extremum are the values of x for which a function f(x) attains a maximum or minimum value.
A(0) = 0
A(3) = 0.002
Therefore, The point of maxima will be x=3, and the maximum area will be 0.002 square units.
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An employment agency requires applicants average at least 70% on a battery of four job skills tests. If an applicant scored 70%, 77%, and 81% on the first three exams, what must he score on the fourth test to maintain a 70% or better average.
Answer:
atleast 52
Step-by-step explanation:
Given that an employment agency requires applicants average at least 70% on a battery of four job skills tests.
An applicant scored 70%, 77%, and 81% on the first three exams,
Since weightages are not given we can assume all exams have equal weights
Let x be the score on the 4th test
Then total of all 4 exams = [tex]70+77+81+x\\= 228+x[/tex]
Average should exceed 70%
i.e.[tex]\bar X \geq 70\\Total\geq 70(4) =280[/tex]
Comparing the two totals we have
[tex]228+x\geq 280\\x\geq 280-228 = 52[/tex]
Hemust score on the fourth test a score atleast 52 to maintain a 70% or better average.
What is the value of x?
Answer:
Step-by-step explanation:
Set this up according to the Triangle Proportionality Theorem:
[tex]\frac{3x}{4x}=\frac{3x+7}{5x-8}[/tex]
Cross multiply to get
[tex]3x(5x-8)=4x(3x+7)[/tex]
and simplify to get
[tex]15x^2-24x=12x^2+28x[/tex]
Get everything on one side of the equals sign and solve for x:
[tex]3x^2-52x=0[/tex] and
[tex]x(3x-52)=0[/tex]
By the Zero Product Property,
x = 0 or 3x - 52 = 0 so x = 17 1/3
An automobile travels past the farmhouse at a speed of v = 45 km/h. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 3.7 km past the intersection of the highway and the road?
Answer:
[tex]\frac{ds}{dt} = 39.586 km/h[/tex]
Step-by-step explanation:
let distance between farmhouse and road is 2 km
From diagram given
p is the distance between road and past the intersection of highway
By using Pythagoras theorem
[tex]s^2 = 2^2 +p^2[/tex]
differentiate wrt t
we get
[tex]\frac{d}{dt} s^2 = \frac{d}{dt} (4 + p^2)[/tex]
[tex]2s \frac{ds}{dt} =2p \frac{dp}{dt} [/tex]
[tex]\frac{ds}{dt} = \frac{p}{s}\frac{dp}{dt}[/tex]
[tex]\frac{ds}{dt} = \frac{p}{\sqrt{p^2 +4}} \frac{dp}{dt}[/tex]
putting p = 3.7 km
[tex]\frac{ds}{dt} = \frac{3.7}{\sqrt{3.7^2 +4}} 45[/tex]
[tex]\frac{ds}{dt} = 39.586 km/h[/tex]
The distance between the automobile and the farmhouse is increasing at the automobile's constant speed of 45 km/h, which is the same as the car's rate of change of distance as it moves away from the farmhouse.
Explanation:The question asks us to determine how fast the distance is increasing between an automobile and a farmhouse when the car is 3.7 km past a certain intersection, given the car's speed is 45 km/h. This is a problem that can be solved using the concepts of rates of change and kinematics.
Given that the car is moving in a straight line away from the farmhouse and there are no other factors altering the speed, the rate at which the distance between the car and farmhouse is increasing is constant and is equal to the speed of the car.
Since the car's speed is constant at 45 km/h, and it moves directly away from the farmhouse without any acceleration or deceleration, the rate at which the distance increases is exactly the car's speed. Therefore, when the car is 3.7 km past the intersection, the distance between the car and the farmhouse is still increasing at 45 km/h.
This straightforward problem shows that when an object moves away from a point at constant speed, the rate at which the distance between the object and the point increases is simply the speed of the object. This concept is very useful in solving more than 100 questions involving rates of change in kinematics, which is a part of classical mechanics.
Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. (5 points)
f(x) = x2 - 3 and g(x) = square root of quantity three plus x
Answer:
f(g(x)) = g(f(x)) = x and f and g are the inverses of each other.
Step-by-step explanation:
Here, the given functions are:
[tex]f(x) = x^2 - 3, g(x) = \sqrt{({3+x)} }[/tex]
To Show: f (g(x)) = g (f (x))
(1) f (g(x))
Here, by the composite function:
[tex]f (g(x)) = f (\sqrt{3+x} ) = \sqrt{(3+x)} ^2 - 3 = (3 + x) - 3 = x[/tex]
⇒ f (g(x)) = x
(2) g (f(x))
Here, by the composite function:
[tex]g(f(x)) = g(x^2 -3) = \sqrt{3 +(x^2 -3) } = \sqrt{x^2} = x[/tex]
⇒ g (f(x)) = x
Hence, f(g(x)) = g(f(x)) = x
⇒ f and g are the inverses of each other.
Mia recently bought a car worth $20,000 on loan with an interest rate of 6.6%. She made a down payment of $1,000 and has to repay the loan within two years (24 months). Calculate her total cost.
Answer:
$22,508
Step-by-step explanation:
Edmentum
The total cost that she pays for car will be $22,508.
What is simple interest?Simple interest is the concept that is used in many companies such as banking, finance, automobile, and so on.
A = P + (PRT)/100
Where P is the principal, R is the rate of interest, and T is the time.
Mia as of late purchased a vehicle worth $20,000 borrowed with a financing cost of 6.6%. She made an initial investment of $1,000 and needs to reimburse the credit in the span of two years (two years).
Then the total cost that she pays will be calculated as,
A = $20,000 + ($19,000 x 6.6 x 2) / 100
A = $20,000 + $2,508
A = $22,508
The total cost that she pays will be $22,508.
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If all men had identical body types, their weight would vary directly as the cube of their height. The tallest person reached a record height of 8 feet 11 inches (107 inches) before his death at age 22. If a man who is 5 feet 10 inches tall (70 inches) with the same body type as the tallest person weighs 170 pounds, what was the tallest person's weight shortly before his death?
Answer: The weight of tallest person would be 607.16 pounds.
Step-by-step explanation:
Since we have given that
height of the tallest person = 8 feet 11 inches = 107 inches
If all men had identical body types, their weight would vary directly as the cube of their height.
If a man who is 5 feet 10 inches tall (70 inches) with the same body type as the tallest person weighs 170 pounds,
So, it becomes,
[tex]W=xh^3[/tex]
[tex]170=x(70)^3\\\\\dfrac{170}{70^3}=x\\\\x=\dfrac{170}{343000}[/tex]
So, weight of tallest person would be
[tex]W=\dfrac{170}{343000}\times 107^3\\\\W=607.16\ pounds[/tex]
Hence, the weight of tallest person would be 607.16 pounds.
First, let's consider the information we're given and set up an equation. We know that, given identical body types, a person's weight would vary directly as the cube of their height. What this means is that the ratio between the weight of two people and the cube of their respective heights will be the same regardless of their individual heights or weights. We can represent this as:
W_tall / W_normal = (Height_tall ^ 3) / (Height_normal ^ 3)
Where:
W_tall is the weight of the tallest person.
W_normal is the weight of the normal person (which we know is 170 pounds).
Height_tall is the height of the tallest person (which we know is 107 inches).
Height_normal is the height of the standard person (which we know is 70 inches).
We are looking to find the weight of the tallest person, so, to isolate W_tall in the equation above, we will multiply both sides of the equation by W_normal:
W_tall = W_normal * (Height_tall ^ 3) / (Height_normal ^ 3)
Substituting the known values into the equation, we get:
W_tall = 170 * (107 ^ 3) / (70 ^ 3)
Computing the values on the right-hand side of the equation, we find that the tallest person weighed approximately 607.16 pounds just before his death.
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The regression equation for predicting number of speeding tickets (Y) from information about driver age (X) is Y = -.065(X) + 5.57. How many tickets would you predict for a twenty-year-old? a. 4 b. 5.57 c. 4.27 d. 6 e. 1
Answer:
c. 4.27
Step-by-step explanation:
We have been given the regression equation of for predicting number of speeding tickets (Y) from information about driver age (X) as [tex]Y=-0.065(X)+5.57[/tex].
To find the predicted number of tickets for a twenty-year-old, we will substitute [tex]X=20[/tex] in our given equation.
[tex]Y=-0.065(20)+5.57[/tex]
[tex]Y=-1.3+5.57[/tex]
[tex]Y=4.27[/tex]
Therefore, the predicted number of tickets for a twenty-year-old would be 4.27 tickets.
You have been saving money in a piggy bank. Your piggy bank contains 75 coins that are all nickels and dimes. You take the money out of the bank to count, and find out that you have $5.95 saved up. How many dimes and how many nickels do you have?
You have 31 nickels and 44 dimes.
Step-by-step explanation:
Total coins = 75
Worth of coins = $5.95 = 5.95*100 = 595 cents
1 nickel = 5 cents
1 dime = 10 cents
Let,
Number of nickels = x
Number of dimes = y
According to given statement;
x+y=75 Eqn 1
5x+10y=595 Eqn 2
Multiplying Eqn 1 by 5
[tex]5(x+y=75)\\5x+5y=375\ \ \ Eqn\ 3\\[/tex]
Subtracting Eqn 3 from Eqn 2
[tex](5x+10y)-(5x+5y)=595-375\\5x+10y-5x-5y=220\\5y=220[/tex]
Dividing both sides by 5
[tex]\frac{5y}{5}=\frac{220}{5}\\y=44[/tex]
Putting y=44 in Eqn 1
[tex]x+44=75\\x=75-44\\x=31[/tex]
You have 31 nickels and 44 dimes.
Keywords: linear equations, subtraction
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In a box of 15 tablets, 4 of the tablets are defective. Three tablets are selected at random. what is the probability that a store buys three tablets and receives: a) no defective tablets, b) one defective tablet, and c) at least one non-defective tablet.
Answer:
a) 0.394
b) 0.430
c) 0.981
Step-by-step explanation:
Use binomial probability:
P = nCr pʳ (1−p)ⁿ⁻ʳ
where n is the number of trials,
r is the number of successes,
and p is the probability of success.
Here, n = 3 and p = 4/15.
r is the number of defective tablets.
a) If r = 0:
P = ₃C₀ (4/15)⁰ (1−4/15)³⁻⁰
P = 1 (1) (11/15)³
P = 0.394
b) If r = 1:
P = ₃C₁ (4/15)¹ (1−4/15)³⁻¹
P = 3 (4/15) (11/15)²
P = 0.430
c) If r ≠ 3:
P = 1 − ₃C₃ (4/15)³ (1−4/15)³⁻³
P = 1 − 1 (4/15)³ (1)
P = 0.981
To find the probability of selecting no defective tablets, multiply the probabilities of selecting non-defective tablets.
Explanation:a) To find the probability of selecting no defective tablets, we need to find the probability of selecting 3 non-defective tablets. There are 11 non-defective tablets out of the total of 15 tablets. So, the probability is:
Multiplying these probabilities together:
(11/15) * (10/14) * (9/13) = 990/2730 ≈ 0.362 = 36.2%
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What is the remainder when 6x+12 is divided by 2x-8.
Answer:
4(3x-2)
Step-by-step explanation
6x+12/2x-8
6x+6x-8
12x-8
4(3x-2)
Answer:
Step-by-step explanation:
2x-8 ) 6x+12 ( 3
6x-24
- +
-----------
36
quotient=3
remainder=36
A recent survey showed that in a sample of 100 elementary school teachers, 15 were single. In a sample of 180 high school teachers, 36 were single. Is the proportion of high school teachers who were single greater than the proportion of elementary teachers who were single? Use α = 0.01.
Answer:
By using hypothesis test at α = 0.01, we cannot conclude that the proportion of high school teachers who were single greater than the proportion of elementary teachers who were single
Step-by-step explanation:
let p1 be the proportion of elementary teachers who were single
let p2 be the proportion of high school teachers who were single
Then, the null and alternative hypotheses are:
[tex]H_{0}[/tex]: p2=p1
[tex]H_{a}[/tex]: p2>p1
We need to calculate the test statistic of the sample proportion for elementary teachers who were single.
It can be calculated as follows:
[tex]\frac{p(s)-p}{\sqrt{\frac{p*(1-p)}{N} } }[/tex] where
p(s) is the sample proportion of high school teachers who were single ([tex]\frac{36}{180} =0.2[/tex])p is the proportion of elementary teachers who were single ([tex]\frac{15}{100} =0.15[/tex])N is the sample size (180)Using the numbers, we get
[tex]\frac{0.2-0.15}{\sqrt{\frac{0.15*0.85}{180} } }[/tex] ≈ 1.88
Using z-table, corresponding P-Value is ≈0.03
Since 0.03>0.01 we fail to reject the null hypothesis. (The result is not significant at α = 0.01)
The proportion of single high school teachers (0.20) is greater than the proportion of single elementary school teachers (0.15). However, further statistical testing would be required to determine if this difference is significant.
Explanation:To answer the student's question regarding proportions, we first need to calculate the proportion of single teachers in both samples. For elementary school teachers, the proportion is 15 out of 100, or 0.15. For high school teachers, the proportion is 36 out of 180 or 0.20.
Now, to determine if the proportion of high school teachers who were single is statistically greater than the proportion of elementary school teachers, we would typically perform a hypothesis test for the difference in proportions. However, in this simplified comparison, we can see that the proportion of single high school teachers (0.20) is indeed greater than the proportion of single elementary school teachers (0.15).
It's important to note that this does not mean there is a significant difference, we would need to conduct a significance test (like a Z-test for two proportions at the α = 0.01 level ) to determine this.
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A sample of 16 ATM transactions shows a mean transaction time of 67 seconds with a sample standard deviation of 12 seconds. State the hypotheses to prove that the mean transaction time exceeds 60 seconds. Assume that times are normally distributed.a. Determine your hypotheses.b. Compute the test statistic. What’s the rejection rule?d. At the α =.05 level of significance, your Critical Value ise. What conclusion can be drawn from this test at a 0.05 significance level?
Answer:
We conclude that the mean transaction time exceeds 60 seconds.
Step-by-step explanation:
We are given the following in the question:
Population mean, μ = 60 seconds
Sample mean, [tex]\bar{x}[/tex] = 67 seconds
Sample size, n = 16
Alpha, α = 0.05
Sample standard deviation, s = 12 seconds
First, we design the null and the alternate hypothesis
[tex]H_{0}: \mu = 60\text{ seconds}\\H_A: \mu > 60\text{ seconds}[/tex]
We use One-tailed(right) t test to perform this hypothesis.
Formula:
[tex]t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}} }[/tex] Putting all the values, we have
[tex]t_{stat} = \displaystyle\frac{67 - 60}{\frac{12}{\sqrt{16}} } = 2.34[/tex]
Now, [tex]t_{critical} \text{ at 0.05 level of significance, 15 degree of freedom } = 1.753[/tex]
Since,
[tex]t_{stat} > t_{critical}[/tex]
We fail to accept the null hypothesis and reject it. Thus, we conclude that the mean transaction time exceeds 60 seconds.
Final answer:
The hypothesis test for the mean ATM transaction time given a sample mean of 67 seconds and a standard deviation of 12 seconds involves a one-tailed Z-test, where the null hypothesis H0: μ = 60 is rejected in favor of the alternative hypothesis Ha: μ > 60 since the test statistic of 2.33 exceeds the critical value of 1.645 at an α = .05 significance level.
Explanation:
Conducting a Hypothesis Test for Mean Transaction Time
For the sample of 16 ATM transactions with a mean of 67 seconds and a standard deviation of 12 seconds, the goal is to test the hypothesis that the mean transaction time exceeds 60 seconds.
a. Determine your hypotheses
The null hypothesis (H0) is that the mean transaction time is 60 seconds (H0: μ = 60). The alternative hypothesis (Ha) is that the mean transaction time is greater than 60 seconds (Ha: μ > 60).
b. Compute the test statistic
To compute the test statistic, we use the following formula for a sample mean with a known standard deviation: Z = (Xbar - μ0) / (s / sqrt(n)) where Xbar is the sample mean, μ0 is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. Plugging in the values we get: Z = (67 - 60) / (12 / sqrt(16)) = 7 / (12 / 4) = 2.33.
c. Rejection rule
The rejection rule is if the computed test statistic is greater than the critical value at α = .05 significance level, we reject H0.
d. Critical Value at α = .05
The critical value for a one-tailed Z-test at α = .05 is approximately 1.645.
e. Conclusion
Because our test statistic of 2.33 exceeds the critical value of 1.645, we reject the null hypothesis, concluding that there is sufficient evidence at the 0.05 significance level to suggest that the mean transaction time exceeds 60 seconds.
Let point M be outside of △ABC. Point N is the reflected image of M about the midpoint of segment AB . Point K is the reflected image of N about the midpoint of segment BC , and point K is the reflected image of L about the midpoint of segment AC . Prove that point A is the midpoint of segment ML .
Explanation:
Define points D, E, F as the midpoints of AB, BC, and AC, respectively. Point D is the midpoint of both AB and MN, so AMBN is a parallelogram, and side AM is parallel to and congruent with side NB.
Point E is the midpoint of both BC and NK, so BNCK is a parallelogram with side NB parallel and congruent to side CK, and by the transitive property of congruence, also to segment AM.
Point F is the midpoint of both AC and KL, so AKCL is a parallelogram with side CK parallel and congruent to side LA. By the transitive properties of congruence and of parallelism, sides AM, NB, CK, and LA are all congruent and parallel. Since AM and LA are congruent to one another and parallel, and share point A, point A must be their midpoint.
please help me!!!!!!!!!!!!!!!!
Answer:
0.91
Step-by-step explanation:
The total of all numbers in the diagram is 35 +5 +10 +5 = 55.
The total of the numbers inside one or both circles is 35 +5 +10 = 50.
The probability of choosing a random student from inside one or both circles (plays some instrument) is 50 out of 55, or ...
50/55 ≈ 0.9090909... ≈ 0.91
P(A∪B) ≈ 0.91
The monthly utility bills in a city are normally distributed with a mean of $121 and a standard deviation of $23. Find the probability that a randomly selected utility bill is between $110 and $130.
To find the probability that a randomly selected utility bill is between $110 and $130, we can use the formula for z-score. By calculating the z-scores for both values, we can find the areas under the curve and subtract them to get the probability.
To find the probability that a randomly selected utility bill is between $110 and $130, we can use the formula for the z-score:
z = (x - μ) / σ
where x is the value we are looking for, μ is the mean, and σ is the standard deviation.
In this case, x = $110, μ = $121, and σ = $23.
Substituting these values into the formula, we get:
z = (110 - 121) / 23 = -0.4783
Using a z-table or a calculator, we can find that the area to the left of -0.4783 is approximately 0.3186.
Next, we repeat the process for $130:
z = (130 - 121) / 23 = 0.3913
Using a z-table or a calculator, we can find that the area to the left of 0.3913 is approximately 0.6480.
To find the probability that the utility bill is between $110 and $130, we subtract the area to the left of $110 from the area to the left of $130:
P(110 ≤ X ≤ 130) = 0.6480 - 0.3186 = 0.3294
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What is the volume of the cylinder?
Answer:
B. 2010.62 ft³
Step-by-step explanation:
The formula for the volume of a cylinder is ...
V = πr²h . . . . . where r is the radius and h is the height
Filling in the numbers and doing the arithmetic, we get ...
V = π(8 ft)²(10 ft) = 640π ft³ ≈ 2010.6193 ft³ ≈ 2010.62 ft³
The volume of the cylinder is about 2010.62 ft³.
Machine A working alone can complete a job in 3 1/2 hours. Machine B working alone can do the same job in 4 2/3 hours. How long will it take both machines working together at their respective constant rates to complete the job?A. 1 hr 10 minB. 2hrC. 4hr 5 minD. 7hrE. 8 hr 10 min
Answer:
B) 2 hours
Step-by-step explanation:
If machine A complete a job in 3 1/2 hours or 7/2 of an hour
means that in one hour finished 1÷ 7/2 or 2/7
If machine B complete a job in 4 2/3 hours or 14/3 of an hour
means that in one hour finished 1÷ 14/3 or 3/14 of an hour
Then the two machines working together in one hour will make
2/7 + 3/14 = (4 + 3)/ 14
or 7/14 = 1/2
half of the job. Therefore these two machines working together will take two hours