Which of the following describes the translation of the graph of y = x 2 to
obtain the graph of y = -x 2 - 3?
reflect over the x-axis and shift left 3
reflect over the x-axis and shift down 3
reflect over the y-axis and shift down 3
Answer:
reflect over the x-axis and shift down 3
Step-by-step explanation:
The leading coefficient of -1 means the graph is reflected over the x-axis. The addition of -3 to the function means each graphed point is shifted down 3 units from the original.
The graph of y = -x^2 -3 is the result of the transformation ...
reflect over the x-axis and shift down 3
Draw the image of quadrilateral ABCD under the translation (x,y)→(x+4,y−3)
Answer:
The overall image would be 4 units to the right and 3 units down on the coordinate plane.
Step-by-step explanation:
We do not see an image. But that is okay. (x+4,y-3) means that the x value of each coordinate gets increased by 4 and the y value of each coordinate gets decreased by 3. The overall image would be 4 units to the right and 3 units down on the coordinate plane.
The image would be 4 units to the right and 3 units down on the coordinate plane.
What is translation?In maths, a translation moves a shape left, right, up, or down but does not turn.
Given the translation of quadrilateral ABCD
(x+4,y-3) means that the x value of each coordinate gets increased by 4 and the y value of each coordinate gets decreased by 3.
Hence, The image would be 4 units to the right and 3 units down on the coordinate plane.
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State the domain and range for the following relation. Then determine whether the relation represents a function.
Father Son
Gem Gale
Hesh Abby
Beni Sam
A. Domain : {Gem, Hesh, Gale}
Range : {Sam, Abby, Beni}
B. Domain : {Gale, Abby, Beni}
Range : {Gem, Hesh, Sam}
C. Domain : {Gem, Hesh, Sam}
Range : {Gale, Abby, Beni}
D. Domain : {Sam, Abby, Beni}
Range : {Gem, Hesh, Gale}
Does the relation represent a function?
A. The relation in the figure is a function because each element in the domain corresponds to exactly one element in the range.
B. The relation in the figure is a function because each element in the range corresponds to exactly one element in the domain.
C. The relation in the figure is not a function because the element Gale in the range corresponds to more than one element in the domain.
D. The relation in the figure is not a function because the element Sam in the domain corresponds to more than one element in the range.
Answer:
C. Domain : {Gem, Hesh, Sam}
Range : {Gale, Abby, Beni}
B. The relation in the figure is a function because each element in the range corresponds to exactly one element in the domain.
Step-by-step explanation:
The figure of the mapping is attached below.
From the diagram, the domain for the relation is the set of Fathers:
{Gem, Hesh, Sam}
The range is the set of Sons:
{Gale, Abby, Beni}
The relation is a function. This is because each element in the range corresponds to exactly one element in the domain.
-8+4(c-9)-5+6c+2c
plz help me out
Answer:
12c-49
Step-by-step explanation:
−8+(4)(c)+(4)(−9)+−5+6c+2c
=−8+4c+−36+−5+6c+2c
Combine Like Terms:
=−8+4c+−36+−5+6c+2c
=(4c+6c+2c)+(−8+−36+−5)
=12c−49
pls mark me brainliest
3 How many ordered pairs of positive integers (a, b) are there such that a right triangle with legs of length a, b has an area of p, where p is a prime number less than 100?
Answer:
The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle.
evaluate the following expressions when x=-4 and y=4
Answer:
1025/4
Step-by-step explanation:
x^6 - x
---------------
4y
Let x =-4 and y = 4
(-4)^6 - (-4)
---------------
4*4
4096 +4
------------------
16
4100/16
1025/4
The answer is C. 1025/4
PLEASE HELP ASAPPPP !!! WILL GIVE BRAINLIEST !!!
1. Find the equation of the line through point (4,−7) and parallel to y=−2/3x+3/2.
2. Find the equation of the line through point (−1,4) and parallel to 5x+y=4
3. Find the equation of the line through point (5,4) and perpendicular to y=−4/3x−2
4. Find the equation of the line through point (8,−9) and perpendicular to 3x+8y=4
Answer:
Step-by-step explanation:
Slope of parallel lines are equal.
1) m = -2/3
(4,-7)
[tex]y-y_{1}=m(x-x_{1})\\\\y-[-7]=\frac{-2}{3}(x-4)\\\\y+7=\frac{-2}{3}*x-\frac{-2}{3}*4\\\\y+7=\frac{-2}{3}x+\frac{8}{3}\\\\y=\frac{-2}{3}x+\frac{8}{3}-7\\\\y=\frac{-2}{3}x+\frac{8}{3}-\frac{7*3}{1*3}\\\\y=\frac{-2}{3}x+\frac{8}{3}-\frac{21}{3}\\\\y=\frac{-2}{3}x-\frac{13}{3}[/tex]
2) 5x + y = 4
y = -5x + 4
slope m = -5
(-1,4)
y - 4 = -5(x -[-1])
y- 4 = -5(x+1)
y - 4 = -5x - 5
y = -5x - 5 + 4
y = -5x -1
If two lines are perpendicular, m2 = -1/m1
m1 = -4/3
[tex]m_{2}=\frac{-1}{\frac{-4}{3}}=-1*\frac{-3}{4}=\frac{3}{4}\\[/tex]
(5,4)
[tex]y-4=\frac{3}{4}(x - 5)\\\\y-4=\frac{3}{4}x-\frac{3}{4}*5\\\\y-4=\frac{3}{4}x-\frac{15}{4}\\\\y=\frac{3}{4}x-\frac{15}{4}+4\\\\y=\frac{3}{4}x-\frac{15}{4}+\frac{4*4}{1*4}\\\\y=\frac{3}{4}x-\frac{15}{4}+\frac{16}{4}\\\\y=\frac{3}{4}x+\frac{1}{4}[/tex]
Answer:
3.) is y=3/4x+1/4 That's what i got but i hope this helps.
Step-by-step explanation:
Takeisha is placing placemats around a table. Each placemat is 14 inches long and 10 inches wide. The rectangular table is 42 inches long and 48 inches wide. What is the greatest number of placemats that Takeisha can place around the table without overlapping?
welll i kinda don get ur question but ummmm if u say like the graetest number then 48
You are thinking about the things that can go wrong on your trip home over the Thanksgiving break. You have booked a flight with US-Scareways. You know that in 33 percent of the cases the company has canceled the flight you were on. Should such a thing occur, there would be no other air travel option home for you. As a backup, your friend Walter has offered you a ride back. However, you know that Walter only has a seat in his car for you with 88 percent probability. What is the probability of you making it home for the holidays?
Answer:
0.9604 = 96.04% probability of you making it home for the holidays
Step-by-step explanation:
We have these following probabilities:
100 - 33 = 67% probability of the flight happening.
In the 33%, that is, when the flight does not happen, there is an 88% probability that you get the ride with Walter.
What is the probability of you making it home for the holidays?
0.67 + 0.88*0.33 = 0.9604
0.9604 = 96.04% probability of you making it home for the holidays
Diego is building a fence for a rectangular garden. It needs to be at least 10 feet wide and at least 8 feet long. The fencing he uses costs $3 per foot. His budget is $120.
Answer:
240
Step-by-step explanation:
8*10=80ft *3= 240 but thats over budget
The total cost to fence the rectangular garden is $108. Diego has enough money to fence the rectangular garden.
What is the perimeter of a rectangle?The perimeter of a rectangle is the total distance of its outer boundary. It is twice the sum of its length and width and it is calculated with the help of the formula: Perimeter = 2(length + width).
Given that, a rectangular garden needs to be at least 10 feet wide and at least 8 feet long.
Now the perimeter of rectangular garden is
2(8+10)
= 36 feet
Total cost of fencing = 36×3
= $108
Diego's budget is $120
Therefore, Diego has enough money to fence the rectangular garden.
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In the previous part, we obtained dy dx = 3t2 − 27 −2t . Next, find the points where the tangent to the curve is horizontal. (Enter your answers as a comma-separated list of ordered pairs.)
x = t^3 - 3t, y = t^2 - 4
Answer:
(27.55, 7.22), (-11.3, 3.21).
Step-by-step explanation:
When is the tangent to the curve horizontal?
The tangent curve is horizontal when the derivative is zero.
The derivative is:
[tex]\frac{dy}{dx} = 3t^{2} - 2t - 27[/tex]
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
[tex]ax^{2} + bx + c, a\neq0[/tex].
This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})[/tex], given by the following formulas:
[tex]x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}[/tex]
[tex]x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}[/tex]
[tex]\bigtriangleup = b^{2} - 4ac[/tex]
In this question:
[tex]3t^{2} - 2t - 27 = 0[/tex]
So
[tex]a = 3, b = -2, c = -27[/tex]
Then
[tex]\bigtriangleup = b^{2} - 4ac = (-2)^{2} - 4*3*(-27) = 328[/tex]
So
[tex]t_{1} = \frac{-(-2) + \sqrt{328}}{2*3} = 3.35[/tex]
[tex]t_{2} = \frac{-(-2) - \sqrt{328}}{2*3} = -2.685[/tex]
Enter your answers as a comma-separated list of ordered pairs.
We found values of t, now we have to replace in the equations for x and y.
t = 3.35
[tex]x = t^{3} - 3t = (3.35)^{3} - 3*3.35 = 27.55[/tex]
[tex]y = t^{2} - 4 = (3.35)^2 - 4 = 7.22[/tex]
The first point is (27.55, 7.22)
t = -2.685
[tex]x = t^{3} - 3t = (-2.685)^3 - 3*(-2.685) = -11.3[/tex]
[tex]y = t^{2} - 4 = (-2.685)^2 - 4 = 3.21[/tex]
The second point is (-11.3, 3.21).
Which represents the reference angle for 3pi/4?
a) 3pi/4
b) Pi/3
c) Pi/4
d) Pi/6
Answer:
(c) Pi/4
Step-by-step explanation:
3pi/4 = (3 x 180) / 4 = 135 degrees
135 degrees is equivalent to (180 degrees - 135 degrees) 45 degrees
45 degrees = Pi/4
Therefore, the reference angle of 3pi/4 is Pi/4
Thus, the correct option is (c) Pi/4
The reference angle for 3π/4 is π/4.
So, the correct answer is c) π/4.
Given is an angle 3π/4, we need to find its reference angle,
The reference angle is the acute angle formed between the terminal side of an angle and the x-axis in the coordinate plane.
To find the reference angle for an angle, you need to consider the position of the terminal side of the angle in the coordinate plane and determine the acute angle formed.
In this case, we are given the angle 3π/4. To find the reference angle, we need to consider the position of the terminal side of this angle.
The angle 3π/4 lies in the second quadrant of the coordinate plane, where both the x-coordinate and y-coordinate are negative.
To find the reference angle, we need to find the acute angle formed by the terminal side with the x-axis. In the second quadrant, this acute angle is formed by the vertical line connecting the terminal side to the x-axis.
If we draw a vertical line from the terminal side of 3π/4, it intersects the x-axis at π/4.
Therefore, the reference angle for 3π/4 is π/4.
So, the correct answer is c) π/4.
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The sum of two rational numbers is
Answer:
rational
Step-by-step explanation:
rational + rational=rational
Expression equivelent to 2/5 divided by 6
Answer:
1/15
Step-by-step explanation:
(2/5)/6 = 2/(5*6) = 2/30 = 1/15
1. Considere as funções f e g, ambas com domínio Z, dadas por f(x) = x²- 2x e g(x) = x³-1. Associe as colunas e assinale a alternativa que apresenta a sequência correta: * 1 ponto Imagem sem legenda a) C, D, A, B b) A, B, D, C c) D, C, A, B d) A, C, D, B
Answer:
(A)C,D,A,B
Step-by-step explanation:
[tex]f(x) = x^2- 2x\\g(x) = x^3-1[/tex]
a) f(-2)
[tex]f(-2) = (-2)^2- 2(-2)\\=4-(-4)\\=4+4\\=8$ (C)[/tex]
b) g(-2)
[tex]g(-2) = (-2)^3-1\\\=-8-1\\=-9 $(D)[/tex]
c) f(-1)+g(3)
[tex]f(-1) = (-1)^2- 2(-1)=1+2=3\\g(3) = (3)^3-1=27-1=26\\f(-1) + g(3)=3+26\\=29$ (A)[/tex]
d) f(5) : g(2)
[tex]f(5) = (5)^2- 2(5)=25-10=15\\g(2) = (2)^3-1=8-1=7\\f(5) :g(2)=15/7$ (B)[/tex]
Answer:
letra A confia
Step-by-step explanation:
The data represents the body mass index (BMI) values for 20 females. Construct a frequency distribution beginning with a lower class limit of 15.0 and use a class width of 6.0. Does the frequency distribution appear to be roughly a normal distribution?
17.7
29.4
19.2
27.5
33.5
25.6
22.1
44.9
26.5
18.3
22.4
32.4
24.9
28.6
37.7
26.1
21.8
21.2
30.7
21.4
Body Mass Index Frequency
15.0 dash 20.9 nothing
21.0 dash 26.9 nothing
27.0 dash 32.9 nothing
Body Mass Index Frequency
33.0 dash 38.9 nothing
39.0 dash 44.9 nothing
Answer:
Given:
Body mass index values:
17.7
29.4
19.2
27.5
33.5
25.6
22.1
44.9
26.5
18.3
22.4
32.4
24.9
28.6
37.7
26.1
21.8
21.2
30.7
21.4
Constructing a frequency distribution beginning with a lower class limit of 15.0 and use a class width of 6.0.
we have:
Body Mass Index____ Frequency
15.0 - 20.9__________3( values of 17.7, 18.3, & 19.2 are within this range)
21.0 to 26.9__________8 values are within this range)
27.0 - 32.9____________ 5 values
33.0 - 38.9____________ 2 values
39.0 - 44.9 _____________2 values
The frequency distribution is not a normal distribution. Here, although the frequencies start from the lowest, increases afterwards and then a decrease is recorded again, it is not normally distributed because it is not symmetric.
The frequency distribution is not a normal distribution because it is not symmetric and this can be obtained through the given data.
Given :
The data represents the body mass index (BMI) values for 20 females.
The frequency distribution begins with a lower class limit of 15.0 and uses a class width of 6.0 are as follows:
Body Mass Index Frequency
15 - 20.9 3
21 - 26.9 8
27 - 32.9 5
33 - 38.9 2
39 - 44.9 2
The frequency distribution is not a normal distribution because it is not symmetric.
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Please help !!!!!!!!!!!!!!!!!!!!!
Answer:
25 and 36
Step-by-step explanation:
5^2=25
6^2=36
Answer:
Im feeling pretty today so imma go with C
Step-by-step explanation:
lol i took the test like about 2 days ago and i just checked my answers and C was right, good luck
*) The scale on a map is 1 : 25000
How many kilometres on the ground is represented by 9 cm on the map?
Answer:
2.25km
Step-by-step explanation:
a scale of 1 : 25000 means:
1 cm on map ------> equals 25000 cm on ground
hence,
9 cm on map ------> equals 25000 x 9 cm = 225,000 cm = 2.25km on ground
What is it ansewerr 1/5 of 45
Answer:
9
Step-by-step explanation:
Please mark me brainliest
The right answer is 9.
please see the attached picture for full solution
Good luck on your assignment
Suppose that daily calorie consumption for american men follows a normal distribution with a mean of 2760 calories and a standard deviation of 500 calories.Suppose a health science researcher selects a random sample of 25 American men and records their calorie intake for 24 hours (1 day). Find the probability that the mean of her sample will be between 2700 and 2800 calories
Answer:
38.11% probability that the mean of her sample will be between 2700 and 2800 calories
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question, we have that:
[tex]\mu = 2760, \sigma = 500, n = 25, s = \frac{500}{\sqrt{25}} = 100[/tex]
Find the probability that the mean of her sample will be between 2700 and 2800 calories
This is the pvalue of Z when X = 2800 subtracted by the pvalue of Z when X = 2700.
X = 2800
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{2800 - 2760}{100}[/tex]
[tex]Z = 0.4[/tex]
[tex]Z = 0.4[/tex] has a pvalue of 0.6554
X = 2700
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{2700 - 2760}{100}[/tex]
[tex]Z = -0.6[/tex]
[tex]Z = -0.6[/tex] has a pvalue of 0.2743
0.6554 - 0.2743 = 0.3811
38.11% probability that the mean of her sample will be between 2700 and 2800 calories
find -34 + 15 - 29 - (-3)
Answer:
-45
Step-by-step explanation:
-34+15=-19
-29-(-3)= -29+3= -26
-19 + -26 = -45
Answer:
-45
Step-by-step explanation:
-34 + 15 - 29 - (-3)
-34-29 +15+3
-63+18
18-63
-45
PLSS I NEED HELP ASAP BECAUSE ITS DUE SOON
Nick has to build a brick wall. Each row of the wall requires 62 bricks. There are 10 rows in the wall. How many bricks will Nick require to build the wall?
A.
102 × 6
B.
106
C.
610
D.
10 × 62
Determine the domain of the function in the correct set notation.
f(x) = 2x+6
O {x|x € R, x*–3}
O {x|x ER, x + 3}
O {x|x € R, x† 2}
O {x|x € R, x*-2}
Answer:
{x|x[tex]\in R[/tex]}
Step-by-step explanation:
We are given that a function
[tex]f(x)=2x+6[/tex]
We have to find the domain of the function in the correct set notation.
The given function is linear polynomial .
The linear polynomial is defined for all real values of x.
Therefore, the given function is defined for values of x.
Domain of f is given by
{x|x[tex]\in R[/tex]}
Hence, the domain of the function in the correct set notation is given by
{x|x[tex]\in R[/tex]}
. a. If 1 adult female is randomly selected, find the probability that her pulse rate is greater than 70 beats per minute. b. If 25 adult females are randomly selected, find the probability that they have pulse rates with a mean greater than 70 beats per minute. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30
Answer:
a) 34.46% probability that her pulse rate is greater than 70 beats per minute.
b) 2.28% probability that they have pulse rates with a mean greater than 70 beats per minute.
c) Because the underlying distribution(female's pulse rate) is normal.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Distribution of females pulse rates:
Here, i suppose there was a typing mistake, since the mean and the standard deviation are lacking.
Also, the question c. only makes sense if the distribution is normal, so i will treat it as being.
I will use [tex]\mu = 68, \sigma = 5[/tex]. I am guessing these values, just using them to explain the question.
a. If 1 adult female is randomly selected, find the probability that her pulse rate is greater than 70 beats per minute.
This is 1 subtracted by the pvalue of Z when X = 70. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{70 - 68}{5}[/tex]
[tex]Z = 0.4[/tex]
[tex]Z = 0.4[/tex] has a pvalue of 0.6554
1 - 0.6554 = 0.3446
34.46% probability that her pulse rate is greater than 70 beats per minute.
b. If 25 adult females are randomly selected, find the probability that they have pulse rates with a mean greater than 70 beats per minute.
Now [tex]n = 25, s = \frac{5}{\sqrt{25}} = 1[/tex]
This probability is 1 subtracted by the pvalue of Z when X = 70. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{70 - 68}{1}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772
1 - 0.9772 = 0.0228
2.28% probability that they have pulse rates with a mean greater than 70 beats per minute.
c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30
The sample size only has to exceed 30 if the underlying distribution is normal. Here, the distribution of females' pulse rate is normal, so this requirement does not apply.
After Clara ran 8 times around a square field , she covered a distance of 288km. Calculate the area of the field.
Answer:
81 km²
Step-by-step explanation:
so the perimeter of the field is 288 : 8 = 36 km
the perimeter of square is 4 time side
4s = 36
s = 36 : 4 = 9 km
Area = side x side = s²
A = 9² = 81 km²
there 2,500 first year students at nicks school. Nick found a brochure from the housing office thats stats 70% of students live on campus. based on the statistics how many students live on campus
Answer:
1750 Students
Step-by-step explanation:
2500*0.7
Use the following information to complete parts (a) through (e) below. A researcher wanted to determine the effectiveness of a new cream in the treatment of warts. She identified 149 individuals who had two warts. She applied cream A on one wart and cream B on the second wart. Test whether the proportion of successes with cream A is different from cream B at the alpha equals 0.05 level of significance. Treatment A Treatment B Success Failure Success 63 10 Failure 25 53 What type of test should be used? A. A hypothesis test regarding the difference of two means using a matched-pairs design. B. A hypothesis test regarding two population standard deviations. C. A hypothesis test regarding the difference between two population proportions from dependent samples. D. A hypothesis test regarding the difference between two population proportions from independent samples.
Answer:
Option D: A hypothesis test regarding the difference between two population proportions from independent samples.
Step-by-step explanation:
A hypothesis test regarding the difference between two population proportions from independent samples. This type of test is used to compare the two population proportions if they are equal or not. It is appropriate under the condition that
The sampling method for each population is simple random sampling.
The samples are independent.
Each sample includes at least 10 successes and 10 failures.
Each population is at least 20 times as big as its sample.
Solve the given inequality. Round to the nearest ten-thousandth, if necessary. e x > 14
Answer:
[tex]x\in(1.146,\infty)[/tex]
Step-by-step explanation:
We are given an inequality
[tex]e^x>14[/tex]
We have to solve the given inequality.
Taking both side ln of given inequality
Then, we get
[tex]ln(e^x)>ln(14)[/tex]
[tex]xlne>1.146[/tex]
We know that
lne=1
Using the value
[tex]x>1.146[/tex]
[tex]x\in(1.146,\infty)[/tex]
Hence, the value of x is given by
[tex]x\in(1.146,\infty)[/tex]
Nuts and bolts are made separately and paired at random. The nut diameters (in mm) are normally distributed with mean 10 and variance .02. The bolt diameters (in mm) are also normally distributed with mean 9.5 and variance 0.02. Find the probability that a bolt is too large for its nut. In other words, find the probability that for a randomly selected nut and bolt, [size of nut − size of bolt] ≤ 0. (Alternatively, you could subtract in the ‘other’ direction and look for the probability that [size of bolt − size of nut] ≥ 0). Draw the distribution of the combination.
Answer:
- The probability that a bolt is too large for its nut = 0.00621
- The image of the drawing of this combined distribution is shown in the attached file to this solution.
Step-by-step explanation:
When independent, normal distributions are combined, the combined mean and combined variance are given through the relation
Combined mean = Σ λᵢμᵢ
(summing all of the distributions in the manner that they are combined)
Combined variance = Σ λᵢ²σᵢ²
(summing all of the distributions in the manner that they are combined)
For this question, the first distribution is the size of a nut, with mean 10 mm and a variance of 0.02
Second distribution is the size of a bolt, with mean 9.5 mm and a variance of 0.02.
The combined distribution is [size of nut − size of bolt]
Hence, λ₁ = 1, λ₂ = -1
μ₁ = 10 mm
μ₂ = 9.5 mm
σ₁² = 0.02
σ₂² = 0.02
Combined Mean = (1×10) + (-1×9.5) = 0.5 mm
Combined Variance = [(1)² × 0.02] + [(-1)² × 0.02] = 0.04
So, the combined distribution is also a normal distribution with a Mean of 0.5 mm and a variance of 0.04.
Standard deviation = √variance = √0.04 = 0.2 mm
The probability that a bolt is too large for its nut, [size of nut − size of bolt] ≤ 0, P(X ≤ 0)
To obtain this required probability, we first normalize/standardize 0 mm
The standardized score for any value is the value minus the mean then divided by the standard deviation.
z = (x - μ)/σ = (0 - 0.5)/0.2 = - 2.50
To determine the required probability
P(x ≤ 0) = P(z ≤ -2.50)
We'll use data from the normal probability table for these probabilities
P(x ≤ 0) = P(z ≤ -2.50) = 0.00621
The image of the drawing of this combined distribution is shown in the attached file to this solution.
Hope this Helps!!!
In a missile-testing program, one random variable of interest is the distance between the point at which the missile lands and the center of the target at which the missile was aimed. If we think of the center of the target as the origin of a coordinate system, we can let Y1 denote the northsouth distance between the landing point and the target center and let Y2 denote the corresponding eastwest distance. (Assume that north and east define positive directions.) The distance between the landing point and the target center is then U=sqrt((y1)^2+(y2)^2). If Y1 and Y2 are independent, standard normal random variables, find the probability density function
Answer:
Step-by-step explanation:
From the given data
we observed that the missile testing program
Y1 and Y2 are variable, they are also independent
We are aware that
[tex](Y_1)^2 and (Y_2)^2[/tex] have [tex]x^2[/tex] distribution with 1 degree of freedom
and [tex]V=(Y_1^2)+(Y_2)^2[/tex] has x^2 with 2 degree of freedom
[tex]F_v(v)=\frac{e^{-\frac{v}{2}}}2[/tex]
Since we have to find the density formula
[tex]U=\sqrt{V}[/tex]
We use method of transformation
[tex]h(V)=\sqrt{U}\\\\=U[/tex]
There inverse function is [tex]h^-^1(U)=U^2[/tex]
We derivate the fuction above with respect to u
[tex]\frac{d}{du} (h^-^1(u))=\frac{d}{du} (u^2)\\\\=2u^2^-^1\\\\=2u[/tex]
Therefore,
[tex]F_v(u)=F_v(h-^1)(u)\frac{dh^-^1}{du} \\\\=\frac{e^-\frac{u^-^}{2} }{2} (2u)\\\\=e^-{\frac{u^2}{2} }U[/tex]