(a)
The probability is : 1/2
(b)
The probability is : 1/2
Step-by-step explanation:The numbers 1, 2, 3, 4, and 5 are written on slips of paper, and 2 slips are drawn at random one at a time without replacement.
The total combinations that are possible are:
(1,2) (1,3) (1,4) (1,5)
(2,1) (2,3) (2,4) (2,5)
(3,1) (3,2) (3,4) (3,5)
(4,1) (4,2) (4,3) (4,5)
(5,1) (5,2) (5,3) (5,4)
i.e. the total outcomes are : 20
(a)
Let A denote the event that the first number is 4.
and B denote the event that the sum is: 9.
Let P denote the probability of an event.
We are asked to find:
P(A|B)
We know that it could be calculated by using the formula:
[tex]P(A|B)=\dfrac{P(A\bigcap B)}{P(B)}[/tex]
Hence, based on the data we have:
[tex]P(A\bigcap B)=\dfrac{1}{20}[/tex]
( Since, out of a total of 20 outcomes there is just one outcome which comes in A∩B and it is: (4,5) )
and
[tex]P(B)=\dfrac{2}{20}[/tex]
( since, there are just two outcomes such that the sum is: 9
(4,5) and (5,4) )
Hence, we have:
[tex]P(A|B)=\dfrac{\dfrac{1}{20}}{\dfrac{2}{20}}\\\\i.e.\\\\P(A|B)=\dfrac{1}{2}[/tex]
(b)
Let A denote the event that the first number is 3.
and B denote the event that the sum is: 8.
Let P denote the probability of an event.
We are asked to find:
P(A|B)
Hence, based on the data we have:
[tex]P(A\bigcap B)=\dfrac{1}{20}[/tex]
( since, the only outcome out of 20 outcomes is: (3,5) )
and
[tex]P(B)=\dfrac{2}{20}[/tex]
( since, there are just two outcomes such that the sum is: 8
(3,5) and (5,3) )
Hence, we have:
[tex]P(A|B)=\dfrac{\dfrac{1}{20}}{\dfrac{2}{20}}\\\\i.e.\\\\P(A|B)=\dfrac{1}{2}[/tex]
Using the probability concept, it is found that:
a) 0.5 = 50% probability that the first number is 4, given that the sum is 9.
b) 0.5 = 50% probability that the first number is 3, given that the sum is 8.
----------------------------------
A probability is the number of desired outcomes divided by the number of total outcomes.The possible outcomes are:
(1,2), (1,3), (1,4), (1,5) .
(2,1), (2,3), (2,4), (2,5).
(3,1), (3,2), (3,4), (3,5).
(4,1), (4,2), (4,3), (4,5).
(5,1), (5,2), (5,3), (5,4).
Item a:
There are 2 outcomes with a sum of 9, which are (4,5) and (5,4).On one of them, (5,4), the first term is 4.Then:
[tex]p = \frac{1}{2} = 0.5[/tex]
0.5 = 50% probability that the first number is 4, given that the sum is 9.
Item b:
There are 2 outcomes with a sum of 8, (3,5) and (5,3).On one of them, (3,5), the first term is 5.Then:
[tex]p = \frac{1}{2} = 0.5[/tex]
0.5 = 50% probability that the first number is 3, given that the sum is 8.
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A flagpole 94.9 ft tall is on the top of a building. From a point on level ground, the angle of elevation of the top of the flagpole is 35.3degrees, while the angle of elevation of the bottom of the flagpole is 26.2degrees. Find the height of the building.
Answer:
214 ft
Step-by-step explanation:
Height of building = 94.9 ft
The angle of elevation of the top of the flagpole = θ₁ = 35.3°
The angle of elevation of the bottom of the flagpole = θ₂ = 26.2°
Let,
Height of building = x
Distance from observation point to base of building = y
[tex]tan 26.2 =\frac{x}{y}\\\Rightarrow y=\frac{x}{tan26.2}[/tex]
[tex]tan 35.3 =\frac{94.9+x}{y}\\\Rightarrow tan 35.3 =\frac{94.9+x}{\frac{x}{tan26.2}}\\\Rightarrow \frac{x}{tan26.2}tan35.3=94.9+x\\\Rightarrow \frac{tan35.3}{tan26.2}x-x=94.9\\\Rightarrow x=\frac{94.9}{\frac{tan35.3}{tan26.2}-1}\\\Rightarrow x=214.84/ ft[/tex]
I have used the exact values from the calculator.
∴ Height of the building is 214.84 ft
Answer:
The height of the building is 214.84 ft.
Step-by-step explanation:
Given information:
The height of the flagpole = 94.9 ft.
The angle of elevation of top = θ[tex]_1[/tex] = [tex]35.3^o[/tex]
The angle of elevation of bottom = θ[tex]_2=26.2^o[/tex]
If the height of building is [tex]x[/tex]
Then,
[tex]tan 26.2=x/y\\y=x/(tan26.2)\\[/tex]
And:
[tex]tan 35.3=(94.9+x)/y\\[/tex]
[tex]94.9+x=y \times tan35.3[/tex]
On putting the value in above equation:
[tex]x=\frac{94.9}{\frac{tan35.3}{tan26.2}-1 }[/tex]
solving the above equation:
[tex]x=214.84 ft.[/tex]
Hence, the height of the building is 214.84 ft.
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Online jewelry sales have increased steadily. In 2003, sales were approximately 2 billion dollars, and in 2013 they were approximately 5.5 1 billion. Construct a model to predict online jewelery sales. Use your model, to find the predicted online jewelry sales for 2015? (Express your answer rounded correctly to the nearest tenth of a billion.)
Answer:
1) [tex]y=2(1+0.081)^x[/tex]
2) 6.4 billion.
Step-by-step explanation:
1) Let the model that is used to find online jewelry sales ( in billions ) after x years,
[tex]y=a(1+r)^x[/tex]
Let 2003 is the initial year,
That is, for 2003, x = 0,
The sales on 2003 is 2 billion, y = 2,
⇒ [tex]2=a(1+r)^0\implies a=2[/tex]
Now, in 2013 they were approximately 5.5 1 billion
i.e. if x = 13 then y = 5.51,
[tex]\implies 5.51 = a(1+r)^{13}[/tex]
[tex]5.51=2(1+r)^{13}[/tex]
With help of graphing calculator,
r = 0.081,
Hence, the model that represents the given scenario is,
[tex]y=2(1+0.081)^x[/tex]
2) For 2015, x = 15,
Hence, online jewelry sales for 2015 would be,
[tex]y=2(1+0.081)^{15}=6.43302745602\approx 6.4\text{ billion}[/tex]
Solve the separable initial value problem. 1. y' = ln(x)(1 + y2), y(1) = 3 = y= tan(xlnx-x+1+arctan(3) 2. y' = 9x? V1 + x? (1 + y2), y(0) = 3 = y=
The solution to the separable initial value problem is [tex]\( y = \tan(x \ln(x) - x + 1 + \arctan(3)) \).[/tex]
Explanation:A differential equation is a mathematical equation that relates a function to its derivatives. It describes how a function's rate of change is related to its current value and possibly other variables. Differential equations are used to model various physical, biological, and social phenomena in fields such as physics, engineering, biology, and economics.
To solve the differential equation [tex]\( y' = \ln(x)(1 + y^2) \)[/tex] with the initial condition [tex]\( y(1) = 3 \)[/tex], we separate variables and integrate both sides.
After integration, we get [tex]\( \tan(y) = x\ln(x) - x + C \)[/tex], where [tex]\( C \)[/tex] is the constant of integration. Using the initial condition, we find [tex]\( C = 1 + \arctan(3) \)[/tex].
Substituting this value back into the equation, we obtain the solution[tex]\( y = \tan(x \ln(x) - x + 1 + \arctan(3)) \).[/tex]This function satisfies the given differential equation and initial condition.
-56 + _ =-84
please help
Answer:
-28 is the answer.
Step-by-step explanation:
84-56=28
56+28=84
-56+-28=-84
Answer:
[tex] - 56 + x = - 84 \\ x = 56 - 84 \\ \boxed{ x = - 28}[/tex]
x= -28 is the right answer.Suppose you first walk 28.6 m in a direction 20 degrees west of north and then 22 m in a direction 40 degrees south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position?
Answer:
29.5 m 64° west of north
Step-by-step explanation:
A suitable vector calculator can add the vectors for you. (See attached.) Here, we have used North as the 0° reference and positive angles in the clockwise direction (as bearings are measured).
___
You can also use a triangle solver (provided by many graphing calculators and stand-alone apps). For this, and for manual calculation (below) it is useful to realize the angle difference between the travel directions is 70°.
___
Using the Law of Cosines to find the distance from start (d), we have (in meters) ...
d² = 28.6² + 22² -2·28.6·22·cos(70°) ≈ 871.561
d ≈ √871.561 ≈ 29.52 . . . . meters
The internal angle between the initial travel direction and the direction to the end point is found using the Law of Sines:
sin(angle)/22 = sin(70°)/29.52
angle = arcsin(22/29.52×sin(70°)) ≈ 44.44°
This angle is the additional angle the destination is west of the initial travel direction, so is ...
20° west of north + 44.44° farther west of north = 64.44° west of north
__
In the second attachment, North is to the right, and West is down. This is essentially a reflection across the line y=x of the usual map directions and angles. Reflection doesn't change lengths or angles, so the computations are valid regardless of how you assign map directions to x-y coordinates.
Find X if AX+B=AC+D
Would it be (D-B/A)+C?
Answer:
it will be x=(AC-B+D)/A
Step-by-step explanation:
AX+B=AC+D
AX+B-B=AC+D-B
AX/A=(AC+D-B)/A
X=(AC+D-B)/A
A recent report in a women magazine stated that the average age for women to marry in the United States is now 25 years of age, and that the standard deviation is assumed to be 3.2 years. A sample of 50 U.S. women is randomly selected. Find the probability that the sample mean age for 50 randomly selected women to marry is at most 24 years
Answer: 0.0136
Step-by-step explanation:
Given : Mean : [tex]\mu=\ 25[/tex]
Standard deviation : [tex]\sigma= 3.2[/tex]
Sample size : [tex]n=50[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x = 24
[tex]z=\dfrac{24-25}{\dfrac{3.2}{\sqrt{50}}}=-2.20970869121\aprox-2.21[/tex]
The P-value =[tex]P(z\leq24)=0.0135526\approx0.0136[/tex]
Hence, the probability that the sample mean age for 50 randomly selected women to marry is at most 24 years = 0.0136
7 = -x+3. Consider the line y = Find the equation of the line that is parallel to this line and passes through the point (-5, 6). Find the equation of the line that is perpendicular to this line and passes through the point (-5, 6)
Answer with explanation:
The equation of line is, y= -x +3
→x+y-3=0---------(1)
⇒Equation of line Parallel to Line , ax +by +c=0 is given by, ax + by +K=0.
Equation of Line Parallel to Line 1 is
x+y+k=0
The Line passes through , (-5,6).
→ -5+6+k=0
→ k+1=0
→k= -1
So, equation of Line Parallel to line 1 is
x+y-1=0
⇒Equation of line Perpendicular to Line , ax +by +c=0 is given by, bx - a y +K=0.
Equation of Line Perpendicular to Line 1 is
x-y+k=0
The Line passes through , (-5,6).
→ -5-6+k=0
→ k-11=0
→k= 11
So, equation of Line Parallel to line 1 is
x-y+11=0
Joyce Meadow pays her three workers $160, $470, and $800, respectively, per week. Calculate what Joyce will pay at the end of the first quarter for (A) state unemployment and (B) federal unemployment. Assume a state rate of 5.6% and a federal rate of .6%. Base is $7,000. A. $950.64; $67.14 B. $655.64; $97.14 C. $755.64; $81.14 D. $850.64; $91.14
Answer:
D. $850.64; $91.14
Step-by-step explanation:
Joyce Meadow pays her three workers $160, $470, and $800, respectively, per week.
So for 1 quarter, he will pay :
[tex]13\times160[/tex] = $2080
[tex]13\times470[/tex] = $6110
[tex]13\times800[/tex] = $10400
Given is that the base is $7,000.
It implies that the unemployment rate is to be paid on the first $7000 only.
Given rates are :
State rate of 5.6% and a federal rate of 0.6%.
State unemployment = [tex](0.056\times2080)+(0.056\times6110)+(0.056\times7000)[/tex] = $850.64
Federal unemployment = [tex](0.006\times2080)+ (0.006\times6110)+ (0.006\times7000)[/tex] = $91.14
So, the correct option is D.
Assume that the ages for first marriages are normally distributed with a mean of 26 years and a standard deviation of 4 years. What is the probability that a person getting married for the first time is in his or her twenties? (Round your answer to 4 decimal places.)
Answer:
0.7743
Step-by-step explanation:
Mean of age = u = 26 years
Standard Deviation = [tex]\sigma[/tex] = 4 years
We need to find the probability that the person getting married is in his or her twenties. This means the age of the person should be between 20 and 30. So, we are to find P( 20 < x < 30), where represents the distribution of age.
Since the data is normally distributed we can use the z distribution to solve this problem. The formula to calculate the z score is:
[tex]z=\frac{x-u}{\sigma}[/tex]
20 converted to z score will be:
[tex]z=\frac{20-26}{4}=-1.5[/tex]
30 converted to z score will be:
[tex]z=\frac{30-26}{4}=1[/tex]
So, now we have to find the probability that the z value lies between -1.5 and 1.
P( 20 < x < 30) = P( -1.5 < z < 1)
P( -1.5 < z < 1 ) = P(z < 1) - P(z<-1.5)
From the z-table:
P(z < 1) = 0.8413
P(z < -1.5) =0.067
So,
P( -1.5 < z < 1 ) = 0.8413 - 0.067 = 0.7743
Thus,
P( 20 < x < 30) = 0.7743
So, we can conclude that the probability that a person getting married for the first time is in his or her twenties is 0.7743
The half-life of Radium-226 is 1590 years. If a sample contains 500 mg, how many mg will remain after 2000 years? Preview mg Give your answer accurate to at least 2 decimal places. Get help: Video Video
Answer:
[tex]a_{n}[/tex]=209.09 mg
Step-by-step explanation:
given: material= radium
half life= 1590 years
initial mass [tex]a_{0}[/tex] =500mg
we know that to calculate the amount left we use
[tex]a_{n}[/tex] = [tex]a_{0}[/tex][tex]\left ( 0.5\right )^{n}[/tex]
[tex]n=\frac{2000}{1590} = 1.2578[/tex]
therefore
[tex]a_{n}[/tex] = [tex]500\times0.5^{1.2578}[/tex]
[tex]a_{n}[/tex]=209.09058407921 mg
[tex]a_{n}[/tex]=209.09 mg amount left after 2000 years
2. Using the Euclidean Algorithm, Calculate the Greatest Common Divisor of the num- bers 11655 and 11340.
Only two steps needed:
11,655 = 1*11,340 + 315
11,340 = 36*315 + 0
This shows that [tex]\mathrm{gcd}(11,655,\,11,340)=315[/tex].
The Greatest Common Divisor (GCD) of the numbers 11655 and 11340 is found using the Euclidean Algorithm by first dividing 11655 by 11340 to get a remainder of 315. This remainder becomes the divisor in the next step and the process repeats until the remainder is zero. Hence, the GCD of 11655 and 11340 is 315.
Explanation:To calculate the Greatest Common Divisor (GCD) of the numbers 11655 and 11340 using the Euclidean Algorithm, follow this process:
Divide 11655 by 11340. The quotient is 1 and the remainder is 315. Now, take the divisor 11340 and divide it by the remainder from the previous step (315). The quotient is 36 with a remainder of 0. The remainder is now zero, which means our last non-zero remainder (315) is the Greatest Common Divisor (GCD).
Therefore, the Greatest Common Divisor of the numbers 11655 and 11340 using the Euclidean Algorithm is 315.
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The price of a sweatshirt at a local shop is twice the price of a pair of shorts.
The price of a T-shirt at the shop is $4 less than the price of a pair of shorts.
Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total
cost of $136.
1. Let w represent the price of one sweatshirt, t represent the price of one Tshirt, and h represent the price of one pair of shorts. Write a system of three
equations that represents the prices of the clothing.
2. Solve the system. Find the cost of each item.
Answer:
The cost of sweatshirt is 12$
Step-by-step explanation:
so we are going to make 3 equations and solve it using the substation method
First- We know the price of the shirt is two times more so w = 2h
Second we know that the the tee shirt is 4 dollars less then the pair of shoes so t = h − 4
Third we take the combined of Brads purchases 3w + 5t + 2h =136
w = 2h
t = h − 4
3w + 2h + 5t = 136
So to solve we are going to to the sub method
3(2h) + 2h + 5(h-4) = 136 - just re written, now we get rid of the ( )
6h + 2h + 5h -20 = 136 - Now we need to move the -20 and add it too 136
6h + 2h + 5h = 156 - Now sum up the H's
13h = 156 - Now divide 13 by 156
h= 12
so the Cost of a sweatshirt is 12 dollars
The cost of a sweatshirt is $24.
The cost of a t shirt is $8
The cost of shorts is $12
Step-by-step explanation:Let w represent the price of one sweatshirt
Let t represent the price of one Tshirt
Let h represent the price of one pair of shorts
The price of a sweatshirt at a local shop is twice the price of a pair of shorts.
[tex]w=2h[/tex] ...(1)
The price of a T-shirt at the shop is $4 less than the price of a pair of shorts.
[tex]t=h-4[/tex] ....(2)
Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total cost of $136.
[tex]3w+2h+5t=136[/tex] .....(3)
Substituting the values of w and t in (3)
[tex]3(2h)+2h+5(h-4)=136[/tex]
=> [tex]6h+2h+5h-20=136[/tex]
=> [tex]13h=136+20[/tex]
=> [tex]13h=156[/tex]
h = 12
t = h-4
[tex]t=12-4=8[/tex]
t = 8
w = 2h
[tex]w=2\times12=24[/tex]
w = 24
-----------------------------------------------------------------------------------
So, the cost of a sweatshirt is $24.
The cost of a t shirt is $8
The cost of shorts is $12
Nadia has 20 more postcards than Pete. After Nadia gives Pete some postcards, Pete has 2 more postcards than Nadia. How many postcards does Nadia give to Pete? 2.
Answer:
11.
Step-by-step explanation:
N = Nadia.
P = Pete.
S = postcards that Nadia gives to Pete.
N = 20 + P
P + S = 2+N-S
To calculate S, we replaces N of the first equation in the second equation:
P + S = 2 + 20 + P-S
2S = 2 + 20 + P - P
2S = 22
S = 22/2 = 11.
6 × 0.9 + 0.1 ÷ 4-0.23
Answer:
5.195
Step-by-step explanation:
To solve the different operations of a mathematical expression there are rules that must always be followed, that is, there is a hierarchy of operations that must be respected.
The hierarchy of operations works as follows:
First, operations in brackets, brackets or braces are resolved independently.
Next, the powers and roots are realized.
It continues to solve multiplications and divisions.
Finally, the addition and subtraction are carried out, regardless of the order.
6 x 0.9 + 0.1 / 4 -0.23 =
First, multiplications and divisions:
5.4 + 0.025 - 0.23.
Then, addition and subraction:
5.195
A company produces a women's bowling ball that is supposed to weigh exactly 14 pounds. Unfortunately, the company has a problem with the variability of the weight. In a sample of 7 of the bowling balls the sample standard deviation was found to be 0.64 pounds. Construct a 95% confidence interval for the variance of the bowling ball weight. Assume normality. a) What is the lower limit of the 95% interval? Give your answer to three decimal places. b) What is the upper limit of the 95% interval? Give your answer to three decimal places. c) Which of the following assumptions should be checked before constructing the above confidence interval? the data need to follow a normal distribution the data need to have small variance the data need to follow a chi-square distribution the data need to be right skewed
Answer:
Step-by-step explanation:
Mean = 14
Std deviation of sample s = 0.64
n = sample size =7
Std error = [tex]\frac{s}{\sqrt{n} } =0.2419[/tex]
t critical for 95% two tailed = 2.02
Margin of error = 2.02*SE = 0.4886
a)Conf interval lower bound = 14-0.4886 = 13.5114
b)Upper bound = 14+0.4886 = 14.4886
c)Assumption
the data need to follow a normal distribution
A process produces a certain part with a mean diameter of 2 inches and a standard deviation of 0.05 inches. The lower and upper engineering specification limits are 1.6 inches and 3.5 inches. What is the Cp (measure of potential capability)?
Answer:
The value of Cp (measure of potential capability) is 6.33.
Step-by-step explanation:
Given information: Process average = 2 inches, process standard deviation = 0.05 inches, lower engineering specification limit = 1.6 inches and upper engineering specification limit =3.5 inches.
The formula for Cp (measure of potential capability) is
[tex]CP=\frac{USL-LSL}{6\sigma}[/tex]
Where, USL is upper specification limit, LSL is specification limit, σ is process standard deviation.
Substitute USL=3.5, LSL=1.6 and σ=0.05 in the above formula.
[tex]CP=\frac{3.5-1.6}{6(0.05)}[/tex]
[tex]CP=\frac{1.9}{0.3}[/tex]
[tex]CP=6.3333[/tex]
[tex]CP\approx 6.33[/tex]
Therefore the value of Cp (measure of potential capability) is 6.33.
suppose the column of the square matrix is linearly independent. What are solutions of Ax=0?
Answer:
the solution is x = 0
Step-by-step explanation:
We know the fact that if the column of the square matrix is linearly independent then the determinant of the matrix is non zero.
Now, since the determinant of the matrix is not zero then the inverse of the matrix must exists.
Therefore, we have
[tex]A^{-1}A=I....(i)[/tex]
Now, for the equation Ax =0
multiplying both sides by [tex]A^{-1}[/tex]
[tex]A^{-1}Ax=A^{-1}\cdot0[/tex]
From equation (i)
[tex]Ix=0\\\\x=0[/tex]
Therefore, the solution is x = 0
A bag of ice pops contains 2 flavored raspberry, 5 flavored lemon, and 3 flavored lime. Find the probability of each event.
a) Picking 2 lemon ice pops at random.
b) picking 2 lime and 1 raspberry at random.
Answer:
A) 2/10 or 20%
B) 3/10 or 30%
Step-by-step explanation:
There are 10 ice pops.
10 is going to be your denominator
Your numerator is going to be whatever amount of ice pops is.
After finding your fraction, divide the numerator by the denominator.
Lastly turn your decimal into a fraction.
2/10=0.2=20%
Hope I helped you.
Use a Taylor Series solution, centered at zero to solve the initial-value problem below. (Find a 5th degree approximation only) dy/dx = x + y y(0) = 1
We're looking for a solution of the form
[tex]y=\displaystyle\sum_{n=0}^\infty a_nx^n=a_0+a_1x+a_2x^2+\cdots[/tex]
Given that [tex]y(0)=1[/tex], we would end up with [tex]a_0=1[/tex].
Its first derivative is
[tex]y'=\displaystyle\sum_{n=0}^\infty na_nx^{n-1}=\sum_{n=1}^\infty na_nx^{n-1}=\sum_{n=0}^\infty(n+1)a_{n+1}x^n[/tex]
The shifting of the index here is useful in the next step. Substituting these series into the ODE gives
[tex]\displaystyle\sum_{n=0}^\infty(n+1)a_{n+1}x^n-\sum_{n=0}^\infty a_nx^n=x[/tex]
Both series start with the same-degree term [tex]x^0[/tex], so we can condense the left side into one series.
[tex]\displaystyle\sum_{n=0}^\infty\bigg((n+1)a_{n+1}-a_n\bigg)x^n=x[/tex]
Pull out the first two terms ([tex]x^0[/tex] and [tex]x^1[/tex]) of the series:
[tex]a_1-a_0+(2a_2-a_1)x+\displaystyle\sum_{n=2}^\infty\bigg((n+1)a_{n+1}-a_n\bigg)x^n=x[/tex]
Matching the coefficients of the [tex]x^0[/tex] and [tex]x^1[/tex] terms on either side tells us that
[tex]\begin{cases}a_1-a_0=0\\2a_2-a_1=1\end{cases}[/tex]
We know that [tex]a_0=1[/tex], so [tex]a_1=1[/tex] and [tex]a_2=1[/tex]. The rest of the coefficients, for [tex]n\ge2[/tex], are given according to the recurrence,
[tex](n+1)a_{n+1}-a_n=0\implies a_{n+1}=\dfrac{a_n}{n+1}[/tex]
so that [tex]a_3=\dfrac{a_2}3=\dfrac13[/tex], [tex]a_4=\dfrac{a_3}4=\dfrac1{12}[/tex], and [tex]a_5=\dfrac{a_4}5=\dfrac1{60}[/tex]. So the 5th degree approximation to the solution to this ODE centered at [tex]x=0[/tex] is
[tex]y\approx1+x+x^2+\dfrac{x^3}3+\dfrac{x^4}{12}+\dfrac{x^5}{60}[/tex]
70% of the workers at Motor Works are female, while 68% of the workers at City Bank are female. If one of these companies is selected at random (assume a 50-50 chance for each), and then a worker is selected at random, what is the probability that the worker will be female?
Answer:
69%
Step-by-step explanation:
If Motor Works is selected, the probability that the worker is a female is 70%. If City Bank is selected, the probability is 68%.
But we don't know what company will be selected, we only know that they have the same probability, 50-50.
So, with 50% of probability the worker will be female with a 70% of probability (because they selected from Motor Works) and with 50% of probability the worker will be female with a 68% of probability (they selected from City Bank).
We express this as 50%*70% + 50%*68% = 69%
The multiplication means that both probabilities happen together and the sum means that happens one thing or the other (they select Motor Works or City Bank)
Final answer:
The probability that a randomly selected worker from either Motor Works or City Bank is female is 0.69, or 69%.
Explanation:
To calculate the probability that a randomly selected worker is female, we need to consider the probability of selecting each company and then the probability of selecting a female worker from that company. We are given that the probability of selecting Motor Works or City Bank is equal, hence it is 1/2 for each. Now, let's calculate the overall probability using the following steps:
Calculate the probability of selecting a female from Motor Works: P(Female at Motor Works) = Probability of Motor Works times Probability of female at Motor Works = 1/2 times 70% = 0.35.
Calculate the probability of selecting a female from City Bank: P(Female at City Bank) = Probability of City Bank times Probability of female at City Bank = 1/2 times 68% = 0.34.
The total probability of selecting a female from either company is the sum of these individual probabilities: P(Female) = P(Female at Motor Works) + P(Female at City Bank) = 0.35 + 0.34 = 0.69.
The probability that a randomly selected worker from either company is female is therefore 0.69, or 69%.
Two fair, distinct dice (one red and one green) are rolled. Let A be the event the red die comes up even and B be the event the sum on the two dice is eight. Are A,B independent events?
Answer: No, A and B are not independent events.
Step-by-step explanation:
Since we have given that
Number of outcomes that a die comes up with = 6
A be the event that the red die comes up even.
A={2,4,6}
B be the event that the sum on the two dice is 8.
B={(2,4),(4,2),(5,3),(3,5),(44)}
P(A) = [tex]\dfrac{3}{6}=\dfrac{1}{2}[/tex]
P(B) = [tex]\dfrac{5}{36}[/tex]
P(A∩B) = [tex]\dfrac{3}{36}[/tex]
But,
P(A).P(B) ≠ P(A∩B)
[tex]\dfrac{1}{2}\times \dfrac{5}{36}\neq \dfrac{3}{36}\\\\\dfrac{5}{72}\neq\dfrac{1}{12}[/tex]
Hence, A and B are dependent events.
Solve the following simultaneous linear congruences.
a) x ? 1 (mod 3), x ? 2 (mod 4), x ? 3 (mod 5).
b) x ? 4 (mod 10), x ? 8 (mod 12), x ? 6 (mod 18).
a. The moduli are coprime, so you can apply the Chinese remainder theorem directly. Let
[tex]x=4\cdot5+3\cdot5+3\cdot4[/tex]
Taken mod 3, the last two terms vanish, and [tex]20\equiv2\pmod3[/tex] so we need to multiply by the inverse of 2 modulo 3 to end up with a remainder of 1. Since [tex]2\cdot2\equiv4\equiv1\pmod3[/tex], we multiply the first term by 2.[tex]x=4\cdot5\cdot2+3\cdot5+3\cdot4[/tex]
Taken mod 4, the first and last terms vanish, and [tex]15\equiv3\pmod4[/tex]. Multiply by the inverse of 3 modulo 4 (which is 3 because [tex]3\cdot3\equiv9\equiv1\pmod4[/tex]), then by 2 to ensure the proper remainder is left.[tex]x=4\cdot5\cdot2+3\cdot5\cdot3\cdot2+3\cdot4[/tex]
Taken mod 5, the first two terms vanish, and [tex]12\equiv2\pmod5[/tex]. Multiply by the inverse of 2 modulo 5 (3, since [tex]3\cdot2\equiv6\equiv1\pmod5[/tex]) and again by 3.[tex]x=4\cdot5\cdot2+3\cdot5\cdot3\cdot2+3\cdot4\cdot3\cdot3[/tex]
[tex]\implies x=238[/tex]
By the CRT, we have
[tex]x\equiv238\pmod{3\cdot4\cdot5}\implies x\equiv-2\pmod{60}\implies\boxed{x\equiv58\pmod{60}}[/tex]
i.e. any number [tex]58+60n[/tex] (where [tex]n[/tex] is an integer) satisifes the system.
b. The moduli are not coprime, so we need to check for possible contradictions. If [tex]x\equiv a\pmod m[/tex] and [tex]x\equiv b\pmod n[/tex], then we need to have [tex]a\equiv b\pmod{\mathrm{gcd}(m,n)}[/tex]. This basically amounts to checking that if [tex]x\equiv a\pmod m[/tex], then we should also have [tex]x\equiv a\pmod{\text{any divisor of }m}[/tex].
[tex]x\equiv4\pmod{10}\implies\begin{cases}x\equiv4\equiv0\pmod2\\x\equiv4\pmod5\end{cases}[/tex]
[tex]x\equiv8\pmod{12}\implies\begin{cases}x\equiv0\pmod2\\x\equiv2\pmod3\end{cases}[/tex]
[tex]x\equiv6\pmod{18}\implies\begin{cases}x\equiv0\pmod2\\x\equiv0\pmod3\end{cases}[/tex]
The last congruence conflicts with the previous one modulo 3, so there is no solution to this system.
Suppose a > 0 is constant and consider the parameteric surface sigma given by r(phi, theta) = a sin(phi) cos(theta)i + a sin(phi) j + a cos(phi) k. 0 lessthanorequalto theta lessthanorequalto 2 pi, 0 lessthanorequalto phi lessthanorequalto pi. (a) Directly verify algebraically that r parameterizes the sphere x^2 + y^2 + z^2 = a^2, by substituting x = a sin(phi), y = a sin(phi) sin(theta), and z = a cos(phi) into the left-hand side of the equation. (b) Find r_phi, r_theta, r_phi times r_theta, and |r_phi times r_theta|. (c) Compute the surface area of the sphere doubleintegral_sigma l dS using change of variables. Find the surface area of the band sigma cut from the paraboloid z = x^2 + y^2 by the planes z = 2 and z = 6 by first finding a parameterization for the surface and then computing doubleintegral_sigma dS. Find the flux of the field F = x^2j - xzk across the surface cut by the parabolic cylinder y = x^2, -1 lessthanorequalto x lessthanorequalto 1, by the planes z = 0 and z = 2, Your normal vector should point in the direction indicated in the figure below.
[tex]\Sigma[/tex] should have parameterization
[tex]\vec r(\varphi,\theta)=a\sin\varphi\cos\theta\,\vec\imath+a\sin\varphi\sin\theta\,\vec\jmath+a\cos\varphi\,\vec k[/tex]
if it's supposed to capture the sphere of radius [tex]a[/tex] centered at the origin. ([tex]\sin\theta[/tex] is missing from the second component)
a. You should substitute [tex]x=a\sin\varphi\cos\theta[/tex] (missing [tex]\cos\theta[/tex] this time...). Then
[tex]x^2+y^2+z^2=(a\sin\varphi\cos\theta)^2+(a\sin\varphi\sin\theta)^2+(a\cos\varphi)^2[/tex]
[tex]x^2+y^2+z^2=a^2\left(\sin^2\varphi\cos^2\theta+\sin^2\varphi\sin^2\theta+\cos^2\varphi\right)[/tex]
[tex]x^2+y^2+z^2=a^2\left(\sin^2\varphi\left(\cos^2\theta+\sin^2\theta\right)+\cos^2\varphi\right)[/tex]
[tex]x^2+y^2+z^2=a^2\left(\sin^2\varphi+\cos^2\varphi\right)[/tex]
[tex]x^2+y^2+z^2=a^2[/tex]
as required.
b. We have
[tex]\vec r_\varphi=a\cos\varphi\cos\theta\,\vec\imath+a\cos\varphi\sin\theta\,\vec\jmath-a\sin\varphi\,\vec k[/tex]
[tex]\vec r_\theta=-a\sin\varphi\sin\theta\,\vec\imath+a\sin\varphi\cos\theta\,\vec\jmath[/tex]
[tex]\vec r_\varphi\times\vec r_\theta=a^2\sin^2\varphi\cos\theta\,\vec\imath+a^2\sin^2\varphi\sin\theta\,\vec\jmath+a^2\cos\varphi\sin\varphi\,\vec k[/tex]
[tex]\|\vec r_\varphi\times\vec r_\theta\|=a^2\sin\varphi[/tex]
c. The surface area of [tex]\Sigma[/tex] is
[tex]\displaystyle\iint_\Sigma\mathrm dS=a^2\int_0^\pi\int_0^{2\pi}\sin\varphi\,\mathrm d\theta\,\mathrm d\varphi[/tex]
You don't need a substitution to compute this. The integration limits are constant, so you can separate the variables to get two integrals. You'd end up with
[tex]\displaystyle\iint_\Sigma\mathrm dS=4\pi a^2[/tex]
# # #
Looks like there's an altogether different question being asked now. Parameterize [tex]\Sigma[/tex] by
[tex]\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+u^2\,\vec k[/tex]
with [tex]\sqrt2\le u\le\sqrt6[/tex] and [tex]0\le v\le2\pi[/tex]. Then
[tex]\|\vec s_u\times\vec s_v\|=u\sqrt{1+4u^2}[/tex]
The surface area of [tex]\Sigma[/tex] is
[tex]\displaystyle\iint_\Sigma\mathrm dS=\int_0^{2\pi}\int_{\sqrt2}^{\sqrt6}u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv[/tex]
The integrand doesn't depend on [tex]v[/tex], so integration with respect to [tex]v[/tex] contributes a factor of [tex]2\pi[/tex]. Substitute [tex]w=1+4u^2[/tex] to get [tex]\mathrm dw=8u\,\mathrm du[/tex]. Then
[tex]\displaystyle\iint_\Sigma\mathrm dS=\frac\pi4\int_9^{25}\sqrt w\,\mathrm dw=\frac{49\pi}3[/tex]
# # #
Looks like yet another different question. No figure was included in your post, so I'll assume the normal vector points outward from the surface, away from the origin.
Parameterize [tex]\Sigma[/tex] by
[tex]\vec t(u,v)=u\,\vec\imath+u^2\,\vec\jmath+v\,\vec k[/tex]
with [tex]-1\le u\le1[/tex] and [tex]0\le v\le 2[/tex]. Take the normal vector to [tex]\Sigma[/tex] to be
[tex]\vec t_u\times\vec t_v=2u\,\vec\imath-\vec\jmath[/tex]
Then the flux of [tex]\vec F[/tex] across [tex]\Sigma[/tex] is
[tex]\displaystyle\iint_\Sigma\vec F\cdot\mathrm d\vec S=\int_0^2\int_{-1}^1(u^2\,\vec\jmath-uv\,\vec k)\cdot(2u\,\vec\imath-\vec\jmath)\,\mathrm du\,\mathrm dv[/tex]
[tex]\displaystyle\iint_\Sigma\vec F\cdot\mathrm d\vec S=-\int_0^2\int_{-1}^1u^2\,\mathrm du\,\mathrm dv[/tex]
[tex]\displaystyle\iint_\Sigma\vec F\cdot\mathrm d\vec S=-2\int_{-1}^1u^2\,\mathrm du=-\frac43[/tex]
If instead the direction is toward the origin, the flux would be positive.
The Royal Fruit Company produces two types of fruit drinks. The first type is 60% pure fruit juice, and the second type is 85% pure fruit juice. The company is attempting to produce a fruit drink that contains 80% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 160 pints of a mixture that is 80% pure fruit juice?
Answer: There are 32 pints of first type and 128 pints of second type in mixture.
Step-by-step explanation:
Since we have given that
Percentage of pure fruit juice in first type = 60%
Percentage of pure fruit juice in second type = 85%
Percentage of pure fruit juice in mixture = 80%
We will use "Mixture and Allegation" to find the ratio of first and second type in mixture:
First type Second type
60% 85%
80%
------------------------------------------------------------------------
85-80 : 80-60
5% : 20%
1 : 4
so, the ratio of first and second type is 1:4.
Total number of pints of mixture = 160
Number of pints of mixture of first type in mixture is given by
[tex]\dfrac{1}{5}\times 160\\\\=32\ pints[/tex]
Number of pints of mixture of second type in mixture is given by
[tex]\dfrac{4}{5}\times 160\\\\=4\times 32\\\\=128\ pints[/tex]
Hence, there are 32 pints of first type and 128 pints of second type in mixture.
amir gupta’s car showroom is giving special offer on one model. their advertised price for this model is four consecutive quarters was $10450 , $10800, $11450, and $9999. use the fourth quarters as a base period. calculate the price index and percentage point rise between each quarters
Answer:
Step-by-step explanation:
S.NO QUATERS PRICE ($) PRICE RELATIVES
[tex]I = \frac{q_i}{q_4} *100[/tex]
1 q_1 10450 104.51
2 q_2 10800 108.01
3 q_3 11450 114.51
4 q_2 9999 100.00
Price Index is given as [tex]= \frac{\sum I}{n}[/tex]
[tex] = \frac{104.51+108.01+114.51+100}{4}[/tex]
= 106.75
b) percentage point rise
[tex]for q_1 = \frac{q_2 -q_1}{q_1}*100[/tex]
[tex]= \frac{108.01-104.51}{104.51}[/tex]
= 3.34%
[tex]for q_2 = \frac{q_3 -q_2}{q_2}[/tex]
[tex]= \frac{114.51-108.01}{108.01} *100[/tex]
= 6.01%
[tex]for q_3 = \frac{q_4 -q_3}{q_3}[/tex]
[tex]= \frac{100-114.51}{114.51} *100[/tex]
= 12.67%
Sample methods that embody random sampling are often termed probability sampling methods. OTrue False
Answer:
The answer is true.
Step-by-step explanation:
Sample methods that embody random sampling are often termed probability sampling methods.
Yes this is a true statement.
Random sampling means picking up the samples randomly from a whole population with each sample having an equal chance of getting selected.
For example- selecting randomly 10 students from each class of a school, to survey for the food quality in school's cafeteria.
And this is a type of probability sampling methods. Other types are stratified sampling, cluster sampling etc.
The U.S. Census Bureau reports that in the year 2008, the mean household income was $68,424 and the median was $50,303. If a histogram were constructed for the incomes of all households in the United States, would you expect it to be skewed to the right, skewed to the left, or approximately symmetric?
Answer:
Skewed to the Right
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced problems.
Since the mean household income is $64,424 and the median was $50,303 then the mean is larger than the median. When this occurs then the constructed histogram is always Skewed to the Right. This is because there are a couple of really large values that affect the mean but not the middle value of the data set.
This in term leads to the graph dipping in values the farther right you go and increasing the farther left you go, as shown in the example picture below.
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Please help if you know the answer...if anyone can show the steps that'd be great
Find the lengths of the sides of the triangle PQR. P(0, 1, 5), Q(2, 3, 4), R(2, −3, 1) |PQ| = Correct: Your answer is correct. |QR| = Correct: Your answer is correct. |RP| = Correct: Your answer is correct. Is it a right triangle? Yes No Is it an isosceles triangle? Yes No
Answer:
1. The values of |PQ|, |QR| and |RP| are 3, 3√5 and 6 respectively.
2. No.
3. No.
Step-by-step explanation:
The vertices of given triangle are P(0, 1, 5), Q(2, 3, 4), R(2, −3, 1).
Distance formula:
[tex]D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}[/tex]
Using distance formula we get
[tex]|PQ|=\sqrt{(2-0)^2+(3-1)^2+(4-5)^2}=\sqrt{9}=3[/tex]
[tex]|QR|=\sqrt{(2-2)^2+(-3-3)^2+(1-4)^2}=\sqrt{45}=3\sqrt{5}[/tex]
[tex]|RP|=\sqrt{(0-2)^2+(1-(-3))^2+(5-1)^2}=\sqrt{36}=6[/tex]
The values of |PQ|, |QR| and |RP| are 3, 3√5 and 6 respectively.
In a right angled triangle the sum of squares of two small sides is equal to the square of third side.
[tex](3)^2+(3\sqrt{5})^2=54\neq 6^2[/tex]
Therefore PQR is not a right angled triangle.
In an isosceles triangle, the length of two sides are equal.
The measure of all sides are different, therefore PQR is not an isosceles triangle.