We are asked to prove the statement,
For all sets A, B, and C, if A ⊆ B then A∪C ⊆ B∪C
Let us consider set A as:
A={1,3,4,5}
and B={1,2,3,4,5,6,7}
Clearly we may observe that A is a subset of B.
( Since, all the elements of set A are contained in set B .
Hence, A is a subset of B)
Now let us consider set C as:
C={1,2}
Hence,
A∪C={1,2,3,4,5}
and
B∪C={1,2,3,4,5,6,7}
Still we observe that:
A∪C ⊆ B∪C
Since all the elements of the set A∪C are contained in the set B∪C.
A Halloween trick-or-treat group consists of 3 Trolls, 4 Nazguls, 4 Ents, and 5 Pokemon. A committee of 4 is to be picked to represent the group at the Monster Bash. (a) Find the probability that the committee will consist of 1 from each type of monster. (b) Suppose that the Pokemon refuse to be on the same committee as a the Trolls. Find the probability that this type of committee is formed.
Answer:
The probability that the committee will consist of 1 from each type of monster is 0.1318 and Suppose that the Pokemon refuse to be on the same committee as a the Trolls.So,the probability that this type of committee is formed is 0.5357
Step-by-step explanation:
No. of Trolls = 3
No. of Nazguls =4
No. of Ents = 4
No. of Pokemon = 5
Total Monsters = 16
We are given that A committee of 4 is to be picked to represent the group at the Monster Bash.
(a) Find the probability that the committee will consist of 1 from each type of monster.
So, the probability that the committee will consist of 1 from each type of monster:
= [tex]\frac{^3C_1 \times ^4C_1 \times ^4C_1 \times ^5C_1}{^{16}C_4}[/tex]
= [tex]\frac{\frac{3!}{1!(3-1)!} \times \frac{4!}{1!(4-1)!} \times \frac{4!}{1!(4-1)!}\times \frac{5!}{1!(5-1)!}}{\frac{16!}{4!(16-4)!}}[/tex]
= [tex]\frac{\frac{3!}{1!(2)!} \times \frac{4!}{1!(3)!} \times \frac{4!}{1!(3)!}\times \frac{5!}{1!(4)!}}{\frac{16!}{4!(12)!}}[/tex]
= [tex]0.1318[/tex]
b)Suppose that the Pokemon refuse to be on the same committee as a the Trolls.
So, the probability that this type of committee is formed
= [tex]\frac{(^3C_3 \times ^8C_1)+(^3C_2 \times ^8C_2)+(^3C_1 \times ^8C_3)+(^3C_0 \times ^8C_4)+(^5C_4 \times ^8C_0)+(^5C_3 \times ^8C_1)+(^5C_2 \times ^8C_2)+(^5C_1 \times ^8C_3)}{^{16}C_4}[/tex]
= [tex]\frac{(\frac{3!}{3!(3-3)!} \times \frac{8!}{1!(8-1)!})+(\frac{3!}{2!(3-2)!} \times\frac{8!}{2!(8-2)!})+(\frac{3!}{1!(3-1)!} \times\frac{8!}{3!(8-3)!})+(\frac{3!}{0!(3-0)!} \times \frac{8!}{4!(8-4)!})+(\frac{5!}{4!(5-4)!} \times \frac{8!}{0!(8-0)!})+(\frac{5!}{3!(5-3)!} \times \frac{8!}{1!(8-1)!})+(\frac{5!}{2!(5-2)!} \times\frac{8!}{2!(8-2)!})+(\frac{5!}{1!(5-1)!} \times \frac{8!}{3!(8-3)!})}{\frac{16!}{4!(16-4)!}}[/tex]
= [tex]0.5357[/tex]
Hence The probability that the committee will consist of 1 from each type of monster is 0.1318 and Suppose that the Pokemon refuse to be on the same committee as a the Trolls.So,the probability that this type of committee is formed is 0.5357
The probability that the committee will consist of 1 member from each type of monster is 1. If the Pokemon refuse to be on the same committee as a Troll, the probability that this type of committee is formed is 4/5 or 0.8.
Explanation:(a) To find the probability that the committee will consist of 1 member from each type of monster, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes. There are 3 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 5 choices for a Pokemon. So the total number of possible outcomes is 3 * 4 * 4 * 5 = 240. Now, we need to calculate the number of favorable outcomes. Since we want 1 member from each type of monster, we choose 1 Troll from 3, 1 Nazgul from 4, 1 Ent from 4, and 1 Pokemon from 5. So the number of favorable outcomes is 3 * 4 * 4 * 5 = 240. Therefore, the probability is 240/240, which simplifies to 1.
(b) If the Pokemon refuse to be on the same committee as a Troll, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes. To do this, we need to consider two cases: one where a Troll is chosen and one where a Troll is not chosen. Case 1: If a Troll is chosen, then there are 3 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 4 choices for a Pokemon (since one Pokemon is excluded). So the total number of outcomes in this case is 3 * 4 * 4 * 4 = 192. Case 2: If a Troll is not chosen, then there are 0 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 5 choices for a Pokemon. So the total number of outcomes in this case is 0 * 4 * 4 * 5 = 0. Therefore, the total number of favorable outcomes is 192 + 0 = 192. The total number of possible outcomes is still 240. So the probability is 192/240, which simplifies to 4/5 or 0.8.
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How many triangles can be made from the following three lengths: 3.1 centimeters, 9.8 centimeters, and 5.2 centimeters?
one
none
more than one
Answer:
none
Step-by-step explanation:
We have three sides
Adding the two smallest sides together must be bigger than the third side
3.1+5.2 > 9.8
8.3> 9.8
This is false, so we cannot make a triangle
When your governor took office, 100,000 children in your state were eligible for Medicaid and 200,000 children were not. Now, thanks to a large expansion in Medicaid, 150,000 children are eligible for Medicaid and 150,000 children are not. Your governor boasts that, under her watch, “the number of children without access to health care fell by one-quarter.” Is this a valid statement to make? Why or why not?
Answer and Step-by-step explanation:
Since we have given that
Number of children in his state were eligible for Medicaid = 100,000
Number of children in his state were not eligible for Medicaid = 200,000
After a large expansion in Medicaid, we get that
Number of children in his state were eligible for Medicaid = 150,000
Number of children in his state were not eligible for Medicaid = 150,000
According to question, , “the number of children without access to health care fell by one-quarter".
So, we check whether it is correct or not.
Difference between previous and current data who were not eligible is given by
[tex]200000-150000\\\\=50000[/tex]
Percentage of decrement is given by
[tex]\dfrac{50000}{200000}=\dfrac{1}{4}[/tex]
Yes , it is fell by one quarter.
A study of king penguins looked for a relationship between how deep the penguins dive to seek food and how long they stay under water. For all but the shallowest dives, there is a linear relationship between depth of dive and length of time under water. The study report gives a scatterplot for a random sample of penguins. The dive duration is measured in minutes and depth (x value) is in meters. The depths are all positive numbers. The dives varied from 40 meters to 300 meters in depth. The report then says, "The regression equation for this study is: y = 2.56 + 0.0135x."
(a) What is the intercept of the regression line? (Use 2 decimal places)
(b) What is the slope of the regression line? (Use 4 decimal places)
(c) What is the correct interpretation of the slope?
For every increase of 1 meter in depth, the mean dive duration increases by exactly 0.0135 minutes.
For every increase of 1 meter in depth, the mean dive duration decreases by exactly 0.0135 minutes.
For every increase of 1 meter in depth, the mean dive duration increases by approximately 0.0135 minutes.
For every increase of 1 meter in depth, the mean dive duration decreases by approximately 0.0135 minutes.
Answer:
given y=2.69+0.0138x the slope is the coefficient of x slope=0.0138 this means that the dive duration is expected to increase by about 0.0138 per minute
Step-by-step explanation:
Answer:
Hi!
A) The intercept of the regression line is 2.56.
B)The slope of the regression line is 0.0135.
C)For every increase of 1-meter depth, the mean dive duration increases by exactly 0.0135 minutes.
Explanation:
The definition of the regression line of a function is the value of Y(x) when x=0.
The slope of a function is the value that configures the ratio of change vertically on the regression.
In this case is represented by 0.0135 multiplied by the x.
If we are talking about depth, and the values of x are all positives values.
Y(50)=2.56+0.0135(50)=3.235
So, when the penguins go deeper, they remain more time diving.
identify the image of XYZ for a composition of a 190 rotation and a 80 rotation, both about point y
Answer:
190° rotation = c
80° rotation = a
Step-by-step explanation:
b = 180° rotation
d = 360° rotation
Answer:
The correct option is c.
Step-by-step explanation:
If the direction of rotation is not mentioned, then it is considered as counterclockwise rotation.
It is given that the figure XYZ rotated 190° and a 80° rotation(composition), both about point y.
It means figure is rotated 80° counterclockwise about the point y after that the new figure is rotated 190° counterclockwise about the point y.
[tex]80^{\circ}+190^{\circ}=270^{\circ}[/tex]
It means the figure XYZ rotated 270° counterclockwise about the point y.
In figure (a), XYZ rotated 90° counterclockwise about the point y.
In figure (b), XYZ rotated 180° counterclockwise about the point y.
In figure (c), XYZ rotated 270° counterclockwise about the point y.
In figure (d), XYZ rotated 360° counterclockwise about the point y.
Therefore the correct option is c.
PLEASE I NEED HELP
Question 13
Write a quadratic function f whose zeros are -6 and -5.
Answer:
The quadratic function is:
[tex]f (x) = x ^ 2 + 11x +30[/tex]
Step-by-step explanation:
The zeros of a function are all values of x for which the function is equal to zero.
If a function has two zeros then it is a quadratic function.
If the zeros are -6 and -5. Then the function will have the following form:
[tex]f (x) = (x + 6) (x + 5)[/tex]
We can expand the expression by applying the distributive property, and we obtain
[tex]f (x) = x ^ 2 + 5x + 6x +30[/tex]
[tex]f (x) = x ^ 2 + 11x +30[/tex]
5. You deposit P1000 into a 9% account today. At the end of two years, you will deposit another P3,000. In five years, you plan a P4000 purchase. How much is left in the account one year after the purchase?
Answer:
Step-by-step explanation:
Since we are talking about compounded annual interest, we can use the Exponential Growth Formula to calculate the answer for this question.
[tex]y = a* (1+r)^{t}[/tex]
Where:
y is the total amount after a given timea is the initial amountr is the interest rate in decimal form t is the amount of timeFirst we need to calculate the total after 2 years with a 9% interest.
[tex]y = 1000* (1+0.09)^{2}[/tex]
[tex]y = 1000* (1.09)^{2}[/tex]
[tex]y = 1000* 1.1881[/tex]
[tex]y = 1188.1[/tex]
So after 2 years there will be £1,188.10 in the account. Now we can add £3000 to that and use the new value as the initial amount, and calculate the new total in 5 years.
[tex]y = (1188.1+3000)* (1+0.09)^{5}[/tex]
[tex]y = 4188.1* (1.09)^{5}[/tex]
[tex]y = 4188.1* 1.5386 [/tex]
[tex]y = 6443.91[/tex]
So now we can subtract the £4000 purchase from the amount currently in the account, and calculate one more year of interest with the new initial amount.
[tex]y = (6443.91-4000)* (1+0.09)^{1}[/tex]
[tex]y = (2443.91)* 1.09[/tex]
[tex]y = 2663.86[/tex]
So at the end you would have £2,662.86 in the account one year after the purchase.
To calculate the amount left in the account one year after the purchase, calculate the interest earned on the initial and subsequent deposits, add them to the account balance, and subtract the purchase amount.
Explanation:To calculate the amount left in the account one year after the purchase, we need to consider the interest earned on the initial deposit and the subsequent deposit in two years. Let's break it down step by step:
Calculate the interest earned on the initial deposit of P1000 over two years:Therefore, there is a deficit of P2010 in the account one year after the purchase.
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Estimate the fur seal pup population in Rookery A 5498 fur seal pups were tagged in early August In late August, a sample of 1300 pups was cbserved and 157 of these were found to have been previously tagged. Use a proportion to estimate the total number of fur seal pups in Rookery A The estimated total number of fur seal pups in Rookery A is Round to the nearest whole number.)
Answer:
the total number of fur seal pups in Rookery A (a) is 45524
Step-by-step explanation:
Given data
in early August (b) = 5498
late August, a sample (c) = 1300
previously tagged (d ) = 157
to find out
the total number of fur seal pups in Rookery A (a)
solution
we will apply here proportion method
that is
a:b :: c :d
a/b = c/d
put all value and find a
a = c/d × b
a = 1300/157 × 5498
a = 45524.84
the total number of fur seal pups in Rookery A (a) is 45524
Given that three fair dice have been tossed and the total of their top faces is found to be divisible by 3, but not divisible by 9. What is the probability that all three of them have landed 4?
The probability of all three dice landing on 4 is 1/216, calculated by multiplying the individual probabilities of each die landing on 4.
Explanation:The probability of all three dice landing on 4 can be calculated using the concept of independent events. Each die has a 1/6 probability of landing on 4, so the probability of all three landing on 4 is (1/6) x (1/6) x (1/6) = 1/216.
Frank Corp has a contribution margin of $450,000 and profit of $150,000. What is its degree of operating leverage?
2.5
3
.33
1.67
Answer:
3
Step-by-step explanation:
given: contribution margin=$450,000 and net profit= $150,000
To find operating leverage
Now, operating leverage is a cost accounting formula that measures the degree to which a firm or project can increase operating income by increasing revenue. A business that generates sales with a high gross margin and low variables costs has high leverage.
[tex]operating leverage=\frac{contribution margin}{net profit}[/tex]
therefore, operating leverage=[tex]\frac{450000}{150000}[/tex] =3
operating leverage is 3
According to a report, 67.5% of murders are committed with a firearm. (a) If 200 murders are randomly selected, how many would we expect to be committed with a firearm? (b) Would it be unusual to observe 153 murders by firearm in a random sample of 200 murders? Why?
Answer: The answer is 135 murders.
Step-by-step explanation: The report tells us that statistically 67.5% of murders are committed using a firearm. It follows therefore that in a sample of 200 randomly selected murders, one would expect that 67.5% of those would be by a firearm. [tex]\frac{67.5}{100}[/tex] * 200 = 135.
It would certainly be higher that the expected value based on previous data collected but it would not be unusual because one sample may have a higher than "normal" amount of murders by firearm. Statistics aren't going to be exact for every sample.
Using the binomial distribution, it is found that:
a) The expected value is of 135.
b) Unusual, as 153 is more than 2.5 standard deviations above the mean.
What is the binomial probability distribution?It is the probability of exactly x successes on n repeated trials, with p probability of a success on each trial.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
In this problem, the parameters are p = 0.675 and n = 200.
Item a:
E(X) = np = 200 x 0.675 = 135.
135 would be expected to be committed with a firearm.
Item b:
The standard deviation is given by:
[tex]\sqrt{V(X)} = \sqrt{200(0.675)(0.325)} = 6.624[/tex]
Then:
[tex]E(X) + 2.5\sqrt{V(X)} = 135 + 2.5(6.624) = 151.56 < 153[/tex]
Since 153 is more than 2.5 standard deviations above the mean, it would be unusual observe 153 murders by firearm in a random sample of 200 murders.
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Use Theorem 7.4.1. THEOREM 7.4.1 Derivatives of Transforms If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then ℒ{tnf(t)} = (−1)n dn dsn F(s). Evaluate the given Laplace transform. (Write your answer as a function of s.) ℒ{te2t sin "3t"}'
Answer:
[tex]L\left(te^{2t }sin3t\right)=\frac{6s-12}{(s^2-4s+13)^2}[/tex].
Step-by-step explanation:
If F(s)= L{f(t)}
Then [tex]L\left\{(t^nf(t)\right\}=(-1)^n\frac{\mathrm{d^n}F(s)}{\mathrm{d^n}s}[/tex]
[tex]L\left\{te^{2t}sin3t\right\}[/tex]
f(t)=[tex]e^{2t}sin3t[/tex]
[tex]L\left\{e^{at}sinbt\right\}=\frac{b}{(s-a)^2+b^2}[/tex]
Therefore,[tex] L\left\{e^{2t}sin3t\right\}=\frac{3}{(s-2)^2+(3)^2}[/tex]
[tex]L\left\{e^{2t}sin3t\right\}=\frac{3}{s^2-4s+13}[/tex]
[tex]L\left\{te^{2t}sin3t\right}=-\frac{\mathrm{d}F(s)}{\mathrm{d}s}[/tex]
=-[tex]\frac{\mathrm{d}e^{2t}sin3t}{\mathrm{d}s}[/tex]
[tex]L\left\{te^{2t}sin3t\right\}[/tex]
[tex]=\frac{3(2s-4)}{(s^2-4s+13)^2}[/tex]
[tex]L\left\{te^{2t}sin3t\right\}=\frac{6s-12}{(s^2-4s+13)^2}[/tex].
The Laplace transform of te2tsin(3t) is 3(-1)(1)/(s2+9)2.
Explanation:According to Theorem 7.4.1, the Laplace transform of a function tnf(t) can be calculated as (−1)n * dn/dsn * F(s), where F(s) is the Laplace transform of f(t). In this case, we need to evaluate the Laplace transform of te2tsin(3t).
First, we need to find the Laplace transform of te2t. Applying the formula F(s) = ℒ{f(t)} = ℒ{te2t}:
F(s) = -d/ds[2/(s-2)2]
Next, we need to find the Laplace transform of sin(3t). Applying the formula F(s) = ℒ{f(t)} = ℒ{sin(3t)}:
F(s) = 3/(s2+9)
Finally, we can evaluate the Laplace transform of te2tsin(3t) as the product of the Laplace transforms obtained:
F(s) = (-1)*(d1/d1s)*(-d1/d1s)[3/(s2+9)]
F(s) = 3(-1)(1)/(s2+9)2
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The hemisphere of radius r is made from a stack of very thin plates such that the density varies with height, r = kz, where k is a constant. Determine its mass and the distance z to the center of mass G.
Answer:
M = ¼ k π R⁴
zG = 8/15 R
Step-by-step explanation:
Note: I'm using lower case r as the radius of each plate and upper case R as the radius of the hemisphere.
The mass of each plate is density times volume:
dm = ρ dV
Each plate has a radius r and a thickness dz. So the volume of each plate is:
dV = π r² dz
Substituting:
dm = ρ π r² dz
We're told that ρ = kz. Substituting:
dm = kz π r² dz
Next, we need to write the radius r in terms of the height z. To do that, we need to look at the cross section (see image below).
The height z and the radius r form a right triangle, where the hypotenuse is the radius of the hemisphere R.
Using Pythagorean theorem:
z² + r² = R²
r² = R² − z²
Substituting:
dm = kπ z (R² − z²) dz
We now have the mass of each plate as a function of its height. To find the total mass, we integrate between z=0 and z=R.
M = ∫ dm
M = ∫₀ᴿ kπ z (R² − z²) dz
M = kπ ∫₀ᴿ (R² z − z³) dz
M = kπ (½ R² z² − ¼ z⁴) |₀ᴿ
M = kπ (½ R⁴ − ¼ R⁴)
M = ¼ k π R⁴
Next, to find the center of gravity, we use the weighted average:
zG = (∫ z dm) / (∫ dm)
zG = (∫ z dm) / M
We already found M, we just have to evaluate the other integral:
∫ z dm
∫₀ᴿ kπ z² (R² − z²) dz
kπ ∫₀ᴿ (R² z² − z⁴) dz
kπ (⅓ R² z³ − ⅕ z⁵) |₀ᴿ
kπ (⅓ R⁵ − ⅕ R⁵)
²/₁₅ k π R⁵
Plugging in:
zG = (²/₁₅ k π R⁵) / (¼ k π R⁴)
zG = ⁸/₁₅ R
Find all the roots of the given function. Use preliminary analysis and graphing to find good initial approximations. f(x)equals=cosine left parenthesis 3 x right parenthesis minus 7 x squared plus 4 xcos(3x)−7x2+4x
Answer:
The given function is
f(x)=cos 3x-7 x²+ 4x
f'(x)=-3 sin 3 x-14 x+4
When you will draw the graph of the function , you will find that root of the function lie between (-1,0).
Consider initial root as,
[tex]x_{0}=0[/tex]
Using Newton method to find the roots of the equation
[tex]x_{n+1}=x_{n}-\frac{f{x_n}}{f'{x_{n}}}\\\\x_{1}=x_{0} - \frac{cos 3x_{0}-7 x_{0}^2+ 4x_{0}}{-3 sin 3 x_{0}-14 x_{0}+4}\\\\x_{1}=-\frac{\cos 0^{\circ}-0+0}{-3 \times 0-0+4}\\\\x_{1}=\frac{-1}{4}\\\\x_{1}= -0.25\\\\x_{2}=x_{1} - \frac{cos 3x_{1}-7 x_{1}^2+ 4x_{1}}{-3 sin 3 x_{1}-14 x_{1}+4}\\\\x_{2}=-0.25 -\frac{cos (-0.75)-7\times (0.0625)- 1}{-3 sin (-0.75)+3.50+4}\\\\x_{2}= -0.176054[/tex]
[tex]x_{3}=x_{2} - \frac{cos 3x_{2}-7 x_{2}^2+ 4x_{2}}{-3 sin 3 x_{2}-14 x_{2}+4}\\\\x_{3}=-0.176054 -\frac{cos (3\times -0.176054)-7\times (-0.176054)^2+4 \times -0.176054}{-3 sin (-0.176054)-14 \times (-0.176054)+4}\\\\x_{3}= -0.1689[/tex]
So, root of the equation is
=0.1688878
=0.1689(approx)
The size of a bicycle is determined by the diameter of the wheel. You want a 20-in bicycle for your birthday. What is the area of one of the wheels? 385.47 in2 314.16 in2 484.37 in2 345.54 in2
Answer:
314.16 in²
Step-by-step explanation:
If the diameter is 20", then the radius is 10" (1/2 the diameter).
Area of a circle is [tex]\pi r[/tex]²
So, π 10² = 100π = 314.16 in²
What is epistemology? Why is it important to critical thinking?
Answer:
Epistemology can be defined as the branch of philosophy which focuses on the theory of knowledge. It is the study of the nature of knowledge, and analyses its relation concepts like truth, justifications and belief. It deals with the rationality of belief(logic or rationale behind a belief).
Yes, it is important to critical thinking as critical thinking is based on beliefs and actions that have reasons, logic and rationale behind them. But these logic and reasons are philosophically problematic as how one evaluate 'reason, 'logic', etc. These are abstract and difficult but question like these are a central part to epistemology.
8) What does the mathematical symbol TT represent? 9) What does the mathematical symbol E represent?
Answer:
TT means pi and e means Euler
Step-by-step explanation:
If 40% of a school consists of boys, 40 % of the students have blonde hair, and three times as many girls as boys have blond hair, what percentage of the school are blonde haired boys?
Answer:
10% of the school are blonde haired boys.
Step-by-step explanation:
Let x be the total number of students,
Number of students who have blonde hair = 40% of the total students
= 40% of x
= 0.4x
Also, the ratio of three times as many girls as boys have blond hair,
So, the ratio of blond hair boys and blond hair girls = 1 : 3
Let the number of boys who have blonde hair = y
And, the number of girls who have blonde hair = 3y
So, y + 3y = 0.4x
4y = 0.4x ⇒ y = 0.1x
Thus, the number of boys who have blonde hair = 0.1x
Hence, the percentage of the school are blonde haired boys
= [tex]\frac{0.1x}{x}\times 100=10\%[/tex]
a bank charges a 1% fee to process a credit card cash advance whis is taken out of the cash advance amount. A customer wants a cash advance of $4,000. how much money will he receive from the bank after the cash advance has been processed.
Answer: [tex]\$\ 3600[/tex]
Step-by-step explanation:
Given : A bank charges a 1% fee to process a credit card cash advance which is taken out of the cash advance amount.
The rate of fee can be also written as 0.01.
The amount of money a customer wants as a cash advance = $4,000
The amount of bank fee = [tex]0.01\times4000=40[/tex]
The amount which customer receive after cash advance has been processed = [tex]4000-40=\$3600[/tex]
Hence, the amount of money will he receive from the bank after the cash advance has been processed = [tex]\$\ 3600[/tex]
Y1=x^4 is a solutionto the ode x^2y"-7xy'+16y=0 use reduction of order to find another independant solution
Answer with explanation:
The given differential equation is
x²y" -7 x y' +1 6 y=0---------(1)
Let, y'=z
y"=z'
[tex]\frac{dy}{dx}=z\\\\y=zx[/tex]
Substitution the value of y, y' and y" in equation (1)
→x²z' -7 x z+16 zx=0
→x² z' + 9 zx=0
→x (x z'+9 z)=0
→x=0 ∧ x z'+9 z=0
[tex]x \frac{dz}{dx}+9 z=0\\\\\frac{dz}{z}=-9 \frac{dx}{x}\\\\ \text{Integrating both sides}\\\\ \log z=-9 \log x+\log K\\\\ \log z+\log x^9=\log K\\\\\log zx^9=\log K\\\\K=zx^9\\\\K=y'x^9\\\\K x^{-9}d x=dy\\\\\text{Integrating both sides}\\\\y=\frac{-K}{8x^8}+m[/tex]
is another independent solution.where m and K are constant of integration.
Answer:
[tex]y_2=x^4lnx[/tex]
Step-by-step explanation:
We are given that a differential equation
[tex]x^2y''-7xy'+16y=0[/tex]
And one solution is [tex]y_1=x^4[/tex]
We have to find the other independent solution by using reduction order method
[tex]y''-\frac{7}{x}y'+\frac{16}{x^2}y=0[/tex]
Compare with the equation
[tex]y''+P(x)y'+Q(x)y=0[/tex]
Then we get P(x)=[tex]-\frac{7}{x}['/tex] Q(x)=[tex]\frac{16}{x^2}[/tex]
[tex]y_2=y_1\int\frac{e^{-\intP(x)dx}}{y^2_1}dx[/tex]
[tex]y_2=x^4\int\frac{e^{\frac{7}{x}}dx}}{x^8}dx[/tex]
[tex]y_2=x^4\int\frac{e^{7ln x}}{x^8}dx[/tex]
[tex]y_2=x^4\int\frac{x^7}{x^8}dx[/tex]
[tex]e^{xlny}=y^x[/tex]
[tex]y_2=x^4\int frac{1}{x}dx[/tex]
[tex]y_2=x^4lnx[/tex]
The Office of Student Services at a large western state university maintains information on the study habits of its full-time students. Their studies indicate that the mean amount of time undergraduate students study per week is 20 hours. The hours studied follows the normal distribution with a standard deviation of six hours. Suppose we select a random sample of 144 current students. What is the probability that the mean of this sample is between 19.25 hours and 21.0 hours?
(A) 0.4332
(B) 0.8664
(C) 0.9104
(D) 0.0181
Answer: (C) 0.9104
Step-by-step explanation:
Given : The hours studied follows the normal distribution
Mean : [tex]\mu=\text{20 hours}[/tex]
Standard deviation : [tex]\sigma=6\text{ hours}[/tex]
Sample size : [tex]n=144[/tex]
The formula to calculate the z-score is given by :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
Let x be the number hours taken by randomly selected undergraduated student.
Then for x = 19.25 , we have
[tex]z=\dfrac{19.25-20}{\dfrac{6}{\sqrt{144}}}=-1.5[/tex]
for x = 21.0 , we have
[tex]z=\dfrac{20-21}{\dfrac{6}{\sqrt{144}}}=2[/tex]
The p-value : [tex]P(19.25<x<21)=P(-1.5<z<2)[/tex]
[tex]P(2)-P(-1.5)= 0.9772498- 0.0668072=0.9104426\approx0.9104[/tex]
Thus, the probability that the mean of this sample is between 19.25 hours and 21.0 hours = 0.9104.
For the equation x^2y" - xy' = 0, find two solutions, show that they are linearly independent and find the general solution. Equations of the form ax^2y" + bxy' + cy = 0 are called Euler's equations or Cauchy-Euler equations. They are solved by trying y = x^r and solving for r (assume that x greaterthanorequalto 0 for simplicity).
Answer:
[tex]y=A+Bx^{2}[/tex]
Step-by-step explanation:
The given Cauchy-Euler equation is: [tex]x^2y''-xy'=0[/tex]
Comparing to the general form: [tex]ax^2y''+bxy'+cy=0[/tex], we have a=1,b=-1 and c=0
The auxiliary solution is given by: [tex]am(m-1)+bm+c=0[/tex]
[tex]\implies m(m-1)-m=0[/tex]
[tex]\implies m(m-1-1)=0[/tex]
[tex]\implies m(m-2)=0[/tex]
[tex]\implies m=0\:\:or\:\:m=2[/tex]
The general solution to this is of the form [tex]y=Ax^{m_1}+Bx^{m_2}[/tex], where A and B are constants.
[tex]y=Ax^{0}+Bx^{2}[/tex]
Therefore the general solution is;
[tex]y=A+Bx^{2}[/tex]
Let [tex]y_1=A[/tex] and [tex]y_2=Bx^2[/tex]
Since we CANNOT express the two solutions as constant multiple of each other, we say the two solutions are linearly independent.
[tex]y_1\neCy_2[/tex], where C is a constant.
The solutions to the differential equation x^2y" - xy' = 0 are y = 1 and y = x, which are linearly independent. The general solution is y = C1 + C2x, where C1 and C2 are constants.
Explanation:For the Cauchy-Euler equation x^2y" - xy' = 0, we find solutions by substituting y = x^r. Differentiating yields y' = rx^{r-1} and y" = r(r-1)x^{r-2}. Substituting these into the equation and simplifying gives us r(r-1)x^r - rx^r = 0, which simplifies to the characteristic equation r^2 - r = 0. Solving this equation, we get the roots r1 = 0 and r2 = 1. Therefore, the two solutions are y1 = x^0 = 1 and y2 = x^1 = x. To show that they are linearly independent, we evaluate the Wronskian determinant W(y1,y2) = |1 x| = x which is nonzero for x > 0. The general solution is a combination of the two: y = C1y1 + C2y2, where C1 and C2 are arbitrary constants.
A pop quiz consists of three true–false questions and three multiple choice questions. Each multiple choice question has five possible answers. If a student blindly guesses the answer to every question, what is the probability that the student will correctly answer all six questions? (Round your answer to 3 decimal places.) Probability
Answer: 0.001
Step-by-step explanation:
Given : Number of true–false questions =3
Choices of answer for true–false questions =2
Then , probability that the answer is correct for a true–false question =[tex]\dfrac{1}{2}=0.5[/tex]
Also, Number of multiple choice questions =3
Choices of answer for multiple choice questions = 5
Then , probability that the answer is correct for a multiple choice question =[tex]\dfrac{1}{5}=0.2[/tex]
Now, the probability that the student will correctly answer all six questions :-
[tex](0.2)^3\times(0.5)^3=0.001[/tex]
Hence, the probability that the student will correctly answer all six questions = 0.001
The probability that a student blindly guessing will correctly answer all six questions on a pop quiz consisting of three true/false questions and three multiple choice questions is 0.001, or 0.1%.
Explanation:The probability of correctly guessing the answer to a true/false question is 0.5, because there are two options, true or false. Therefore, the probability of correctly guessing all three true/false answers is 0.5^3, which is 0.125.
For a multiple choice question with five answers, the probability of guessing correctly is 0.2 (or 1/5). Therefore, the probability of correctly answering all three multiple choice questions is 0.2^3, which is 0.008.
To find the total probability of correctly answering all six questions, we multiply these two probabilities together: 0.125 * 0.008 = 0.001. So, the probability that a student guessing blindly will correctly answer all six questions is 0.001, or 0.1% when expressed as a percentage.
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Let R be the relation on N x N defined by (a, b) R(c, d) if and only if ad bc. Show that R Equivalence Relations. is an equivalence relation on N x N.
Last year, a person wrote 123 checks. Let the random variable x represent the number of checks he wrote in one day, and assume that it has a Poisson distribution. What is the mean number of checks written per day? What is the standard deviation? What is the variance?
Answer:The mean number of checks written per day = 0.3370
The standard deviation = 0.5805
The variance = 0.3370
Step-by-step explanation:
Let the random variable x represent the number of checks he wrote in one day.
Given : The number of checks written in last year = 123
Let the number of days in the year must be 365.
Now, the mean number of checks written per day will be :-
[tex]\lambda=\dfrac{123}{365}=0.33698630137\approx0.3370[/tex]
We know that in Poisson distribution , the variance is equals to the mean value .
[tex]\text{Thus , Variance }=\sigma^2= 0.3370[/tex]
[tex]\Rightarrow\ \sigma=\sqrt{0.3370}=0.580517010948\approx0.5805[/tex]
Thus, Standard deviation = 0.5805
Compute the following quantities: a) i^21-i^32
b) 2-i/3-2i.
Please show work
Answer: a) i-1; b) 4+i/13
Step-by-step explanation:
The complex number [tex]i[/tex] is defined as the number such that [tex]i^{2}=-1[/tex];
We use the propperty to notice that [tex]i^1=i \quad i^2 = -1 \quad i^3=-i \quad i^4=1 \quad i^5=1 \quad i^6=-1 \quad i^7=-i \quad i^8=1 \quad i^9=i \quad i^10= -1 etc...[/tex].
a) We notice that [tex]i^{21}=i \quad \text{ and \text} \quad i^{32}=1[/tex]. Hence, [tex]i^{21}-i^{32}=i-1[/tex].
b) We multiply the expression by [tex]1=\frac{3+2\cdot i}{3 + 2 \cdot i}[/tex]. Then we get that
[tex]\frac{2-i}{3-2 \cdot i}=\frac{2-i}{3-2\cdot i}\cdot\frac{3+2 \cdot i}{3 + 2\cdot i } = \frac{(2-i)\cdot(3+2 \cdot i)}{3^2+2^2}= \frac{6+4i-3i+2i^2}{13}=\frac{6+i-2}{13} = \frac{4+i}{13}[/tex]
Please someone help me with these equations
Answer:
[tex]f (-2) =-\frac{8}{3}[/tex]
[tex]f (4) =\frac{4}{3}[/tex]
[tex]f (1) = - 4[/tex]
Step-by-step explanation:
For this case it has a piecewise function composed of two functions.
To evaluate the piecewise function observe the condition.
[tex]f (x) = \frac{1}{3}x ^ 2 -4[/tex] when [tex]x \neq 1[/tex]
[tex]f (x) = -4[/tex] when [tex]x = 1[/tex]
We start by evaluating [tex]f(-2)[/tex], note that [tex]x = -2\neq 1[/tex]. Then we use the quadratic function:
[tex]f (-2) = \frac{1}{3}(-2) ^ 2 -4 = -\frac{8}{3}[/tex]
Now we evaluate [tex]f(4)[/tex] note that [tex]x = 4\neq 1[/tex]. Then we use the quadratic function:
[tex]f (4) = \frac{1}{3}(4) ^ 2 -4 = \frac{4}{3}[/tex]
Finally we evaluate [tex]f(1)[/tex] As [tex]x = 1[/tex] then
[tex]f (1) = - 4[/tex]
1. Using the Euclidian algorithm, compute (91,39) and (73,21)
Answer:
HCF(91,39) = 13 and HCF(73,21) = 1
Step-by-step explanation:
As per euclidian algorithm, a = bq + r, where a is dividend, b is divisor, q is quotient and r is remainder.
We can use euclidian algorithm to find the HCF of numbers.
To find: HCF ( 91, 39 ):
On dividing 91 by 39, we get
91=39×2+13
Here, remainder = 13 [tex]\neq 0[/tex]
So, again applying division algorithm on 39 and 13, we get
[tex]39=13\times 3+0[/tex]
As remainder = 0 and divisor at this step is equal to 13, HCF = 13 .
To find: HCF ( 73, 21 )
On dividing 73 by 21, we get
[tex]73=21\times 3+10[/tex]
Here, remainder = 10 [tex]\neq 0[/tex]
On applying division algorithm on 21 and 10, we get
[tex]21=10\times 2+1[/tex]
Here, remainder = 1 [tex]\neq 0[/tex]
On applying division algorithm on 10 and 1, we get
[tex]10=1\times 10+0[/tex]
As remainder = 0 and divisor at this step is 1, HCF = 1
A tank initially holds 100 gallons of a brine solution containing 1 lb. of salt. At t = 0 another brine solution containing 1 pound of salt per gallon is poured into the tank at the rate of 4 gal/min., while the well-stirred mixture leaves the tank at the same rate. Find (a) the amount of salt in the tank at any time t and (b) the time at which the mixture in the tank contains 2 lb of salt.
Answer:
[tex]\boxed{\text{(a)}A = 100 - 99e^{-t/25}; \,\text{(b) 15 s}}[/tex]
Step-by-step explanation:
(a) Expression for mass of salt as function of time
[tex]\text{Let A = mass of salt after t min}\\\text{and }r_{i} = \text{rate of salt coming into talk}\\\text{and }r_{o}$ =\text{rate of salt going out of tank}[/tex]
i. Set up an expression for the rate of change of salt concentration.
[tex]\dfrac{\text{d}A}{\text{d}t} = r_{i} - r_{o}\\\\r_{i} = \dfrac{\text{4 gal}}{\text{1 min}} \times \dfrac{\text{1 lb}}{\text{1 gal}} = \text{4 lb/min}\\\\r_{o} = \dfrac{\text{4 gal}}{\text{1 min}} \times \dfrac {A\text{ lb}}{\text{100 gal}} =\dfrac{A}{25}\text{ lb/min}\\\\\dfrac{\text{d}A}{\text{d}t} = 4 - \dfrac{x}{25}[/tex]
ii. Integrate the expression
[tex]\dfrac{\text{d}A}{\text{d}t} = \dfrac{100 - A}{25}\\\\\dfrac{\text{d}A}{100 - A} = \dfrac{\text{d}t}{25}\\\\\int \frac{\text{d}A}{100 - A} = \int \frac{\text{d}t}{25}\\\\-\ln(100 - A) = \dfrac{t}{25} + C[/tex]
iii. Find the constant of integration
[tex]-\ln (100 - A) = \dfrac{t}{25} + C\\\\\text{At $t$ = 0, $A$ = 1, so}\\\\-\ln (100 - 1) = \dfrac{0}{25} + C\\\\C = -\ln 99[/tex]
iv. Solve for A as a function of time.
[tex]\text{The integrated rate expression is}\\\\-\ln (100-A) = \dfrac{t}{25} - \ln 99\\\\\text{Solve for } A\\\\\ln(100 - A) = \ln 99 - \dfrac{t}{25}\\\\100 - A = 99e^{-t/25}\\\\A = \boxed{\mathbf{100 - 99e^{-t/25}}}[/tex]
The diagram shows A as a function of time. The mass of salt in the tank starts at 1 lb and increases asymptotically to 100 lb.
(b) Time to 2 lb salt
[tex]A = 100 - 99e^{-t/25}\\\\2 = 100 - 99e^{-t/25}\\\\99e^{-t/25} = 98\\\\e^{-t/25} = 0.989899\\\\-t/25 = -0.01015\\\\t = 25\times 0.01015 =\text{0.25 min = 15 s}\\\\\text{The tank will contain 2 lb of salt after } \boxed{\textbf{15 s}}[/tex]
To find the amount of salt in the tank at any time t, we need to consider the rate at which the brine solution is being poured in and leaving the tank. Initially, there is 1 pound of salt in the tank. The rate at which salt is added is 1 pound per gallon, and the rate at which the mixture is being removed is 4 gallons per minute. Therefore, the amount of salt in the tank at time t can be calculated using the equation. To find the time at which the mixture in the tank contains 2 pounds of salt, we can set up the equation. Solving this equation for t will give us the time at which the mixture contains 2 pounds of salt.
Explanation:To find the amount of salt in the tank at any time t, we need to consider the rate at which the brine solution is being poured in and leaving the tank. Initially, there is 1 pound of salt in the tank. The rate at which salt is added is 1 pound per gallon, and the rate at which the mixture is being removed is 4 gallons per minute. Therefore, the amount of salt in the tank at time t can be calculated using the equation:
Amount of salt at time t = 1 + (1 pound/gallon)(4 gallons/minute)(t minutes)
To find the time at which the mixture in the tank contains 2 pounds of salt, we can set up the equation:
2 = 1 + (1 pound/gallon)(4 gallons/minute)(t minutes)
Solving this equation for t will give us the time at which the mixture contains 2 pounds of salt.
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A worker is cutting a square from a piece of sheet metal. The specifications call for an area that is 16 cm squared with an error of no more than 0.03 cm squared. How much error could be tolerated in the length of each side to ensure that the area is within the tolerance?
Given:
area of square, A = 16 [tex]cm^{2}[/tex]
error in area, dA = 0.03 cm^{2}
Step-by-Step Explanation:
Let 'a' be the side of the square
area of square, A = [tex]a^{2}[/tex] (1)
A = 16 = [tex]a^{2}[/tex]
Therefore, a = 4 cm
for max tolerable error in length 'da', differentiate eqn (1) w.r.t 'a':
dA = 2a da
[tex]0.03 = 2\times 4\times da[/tex]
da = [tex]\frac{0.03}{8}[/tex]
da = 0.0375 cm
The side length of the square is 4 cm, the maximum error that can be tolerated in the length of each side to ensure that the area is within the specified tolerance is 0.0375 cm (or 0.0375 mm).
Given specifications:
Desired Area (A) = 16 cm²
Tolerance (ΔA) = 0.03 cm²
The formula for the area of a square is:
A = side length (L) * side length
Calculate the derivative of the area formula with respect to the side length (L):
dA/dL = 2L
Now, we want to find the maximum error in the side length (ΔL) that can be tolerated while keeping the area within the specified tolerance:
ΔA = (dA/dL) * ΔL
Plug in the values we have:
0.03 cm² = (2L) * ΔL
Solve for ΔL:
ΔL = 0.03 cm² / (2L)
To ensure that the area is within the tolerance, the error in the side length should be no more than ΔL.
Now, let's calculate ΔL using the formula above and for a given side length (L):
ΔL = 0.03 cm² / (2L)
If we assume a side length of L = 4 cm (to achieve the desired area of 16 cm²), we can calculate ΔL:
ΔL = 0.03 cm² / (2 * 4 cm) = 0.03 cm² / 8 cm = 0.00375 cm = 0.0375 mm
So, if the side length of the square is 4 cm, the maximum error that can be tolerated in the length of each side to ensure that the area is within the specified tolerance is 0.0375 cm (or 0.0375 mm).
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