Check the Wronskian determinant:
[tex]W(e^{x/2},xe^{x/2})=\begin{vmatrix}e^{x/2}&xe^{x/2}\\\frac12e^{x/2}&\left(1+\frac x2\right)e^{x/2}\end{vmatrix}=\left(1+\frac x2\right)e^x-\frac x2e^x=e^x\neq0[/tex]
The determinant is not zero, so the solutions are indeed linearly independent.
To verify a fundamental set of solutions for the given differential equation, one must demonstrate that the functions e^{x/2} and xe^{x/2} satisfy the equation and that their Wronskian is non-zero, indicating linear independence.
Explanation:The student's question pertains to verifying whether a given set of functions, e^{x/2} and xe^{x/2}, form a fundamental set of solutions for the differential equation 4y'' - 4y' + y = 0. A set of solutions is fundamental if the functions are linearly independent and satisfy the differential equation. Linear independence can be proved by calculating the Wronskian, which must be non-zero over the given interval. To show that these functions are solutions, they must be substituted into the differential equation to check if it holds true.
To check for linear independence, we can compute the Wronskian:
W(e^{x/2}, xe^{x/2}) = |which simplifies to e^{x} (1 - (x/2)) that is non-zero for all real numbers x, proving linear independence.
To verify if the functions satisfy the differential equation, we substitute each function into the equation. The derivatives of e^{x/2} and xe^{x/2} are taken, and then these are plugged into the equation to confirm that it yields zero.
In 2005, there were 18,400 students at college A, with a projected enrollment increase of 500 students per year In the same year, there were 37,650 students at college B, with a projected enrollment dec line of 1250 students per year According to these projections, when will the colleges have the same enrollment? What will be the enrolliment in each college at that time? In the yearthe enrolment at both colleges will be the same The total enrolment at each college will bestudents
Answer:
all parts has been answered
Step-by-step explanation:
Let us assume
after T years both colleges have same enrollment
Enrollment at college A after T years = 18400+500*T
Enrollment at college B after T years = 37650-1250*T
Both the colleges will have same enrollment after T years
Hence
18400+500*T=37650-1250*T
1750*T=19250
T=11 years
Present year would be =2005+11=2016
In the year 2016, the enrollment of both the colleges will be same.
Total enrollment at each college = 18400+11*500=37650-1250*11=23900
The total enrollment at each college will be 23900 students
Find a possible formula for a fourth degree polynomial g that has a double zero at -2, g(4) = 0, g(3) = 0, and g(0) = 12. g(x) =
The possible formula for a fourth degree polynomial g is:
[tex]g(x)=\dfrac{1}{4}(x^4-3x^3-12x^2+20x+48)[/tex]
Step-by-step explanation:We know that if a polynomial has zeros as a,b,c and d then the possible polynomial form is given by:
[tex]f(x)=m(x-a)(x-b)(x-c)(x-d)[/tex]
Here the polynomial g has a double zero at -2, g(4) = 0, g(3) = 0.
This means that the polynomial g(x) is given by:
[tex]g(x)=m(x-(-2))^2(x-4)(x-3)\\\\i.e.\\\\g(x)=m(x+2)^2(x-4)(x-3)\\\\i.e.\\\\g(x)=m(x^2+2^2+2\times 2\times x)(x(x-3)-4(x-3))\\\\i.e.\\\\g(x)=m(x^2+4+4x)(x^2-3x-4x+12)\\\\i.e.\\\\g(x)=m(x^2+4+4x)(x^2-7x+12)\\\\i.e.\\\\g(x)=m[x^2(x^2-7x+12)+4(x^2-7x+12)+4x(x^2-7x+12)]\\\\i.e.\\\\g(x)=m[x^4-7x^3+12x^2+4x^2-28x+48+4x^3-28x^2+48x]\\\\i.e.\\\\g(x)=m[x^4-7x^3+4x^3+12x^2+4x^2-28x^2-28x+48x+48]\\\\i.e.\\\\g(x)=m[x^4-3x^3-12x^2+20x+48][/tex]
Also,
[tex]g(0)=12[/tex]
i.e.
[tex]48m=12\\\\i.e.\\\\m=\dfrac{12}{48}\\\\i.e.\\\\m=\dfrac{1}{4}[/tex]
Hence, the polynomial g(x) is given by:
[tex]g(x)=\dfrac{1}{4}(x^4-3x^3-12x^2+20x+48)[/tex]
The formula for the fourth degree polynomial g(x) can be determined using the known roots and points. The polynomial will have form g(x) = a(x+2)² × (x-3) × (x-4). Value of 'a' is found by setting x=0 and solving for a.
Explanation:The subject of the question is determining the formula for a fourth degree polynomial, g(x), based on provided conditions. From the question we know that g(x) must have a double root at -2 (-2, -2); roots 3 and 4 (3, 0) and (4, 0); and that g(0) = 12.
With this information, we know that a fourth degree polynomial will look something like this: g(x) = ax4 + bx3 + cx2 + d = 0. But, per the conditions, it looks like g(x) = a(x+2)2 × (x-3) × (x-4), because it has roots at x = -2 (twice, or 'double'), x = 3, and x = 4.
To find the value of 'a', we use the information that at x = 0, g(x) should equal to 12. The function can then be processed as follows:
g(0) = a(0+2)2 × (0-3) × (0-4) = 12.
This will give us 'a', and the desired polynomial formula.g(x).
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The scores on the entrance exam at a well-known, exclusive law school are normally distributed with a mean score of 200 and a standard deviation equal to 50. At what value should the lowest passing score be set if the school wishes only 2.5 percent of those taking the test to pass? (Round your answer to nearest whole number.)
Answer:
the lowest passing score would be x = 298
Step-by-step explanation:
School wishes that only 2.5 percent of students taking test pass
We are given
mean= 200,
standard deviation = 50
We need to find x
The area under the curve can be found by:
2.5 % = 0.025
So, 1- 0.025 = 0.975
We need to find the value of z for which the answer is 0.975
Looking at the z-score table, the value of z is: 1.96
Now, using the formula:
z = x - mean/standard deviation
1.96 = x - 200/50
=> 1.96 * 50 = x-200
98 = x - 200
=> x = 200+98
x = 298
So, the lowest passing score would be x = 298
The lowest passing score should be set at 102 to ensure that only 2.5 percent of the test takers pass.
Understand the Problem Context:Voting age
17-29 30-44 45-64 65+
9 8 32 15
What is the probability that a voter is younger than 45?
Answer:
[tex]\frac{17}{64}\approx 0.27[/tex]
Step-by-step explanation:
We have been given a table to voters and their ages. We are asked to find the probability that a voter is younger than 45.
Voting age Voters
17-29 9
30-44 8
45-64 32
65+ 15
We can see from our given table that age of 17 (9+8) voters is between 17 to 44 years.
To find the probability that a voter is younger than 45, we will divide 17 by total number of voters.
[tex]\text{Total voters}=9+8+32+15=64[/tex]
[tex]\text{Probability that a voter is younger than 45}=\frac{17}{64}[/tex]
[tex]\text{Probability that a voter is younger than 45}=0.265625[/tex]
[tex]\text{Probability that a voter is younger than 45}\approx 0.27[/tex]
Therefore, the probability that a voter is younger than 45 is 0.27.
Find the density in lbs/cbf, round to nearest tenth...... please urgent request i have 30 minutes left
180 pounds; 15” x 15” x 20” __________________________ lbs/cbf
150 cf; 90 kg = _______________________________ lbs/cbf
Answer:
d1=69.12 lbs/cbf, d2=1.32 lbs/cbf
Step-by-step explanation:
Hello
to make the conversion we will need
1" = 1 inch
12 inch = 1 feet
1 kg= 2. 20 lbs
Point 1, step 1
convert inch to feet
[tex]15"=15 inch*(\frac{1 feet}{12 in})=\frac{5}{4} ft\\ 20"=20 inch*(\frac{1 feet}{12 in})=\frac{5}{3}ft\\d=\frac{m(lbs)}{v(cbf)}\\ d=\frac{180 lbs}{\frac{5}{4} ft*\frac{5}{4} ft*\frac{5}{3} ft}\\ d=69.12\ lbs/cbf[/tex]
Point 2, step 2
[tex]90kg=90kg*\frac{2.2 lbs}{1 kg} =198 lbs\\\\d=\frac{m}{v}\\ d=\frac{198 lbs}{150 cbf}\\d=1.32\ lbs/cbf[/tex]
I hope it helps
Suppose that you invest S1,100 in stock. Four years later, your investment yields $1,775. What is the rate of return of your investment? The rate of return is %. (Round to one decimal place.)
Answer:
The rate of return is 61.3%.
Step-by-step explanation:
Rate of return is given by:
[tex]\frac{current price - original price}{original price}\times100[/tex]
= [tex]\frac{1775-1100}{1100} \times100[/tex]
= [tex]\frac{675}{1100} \times100[/tex]
= 61.36% ≈ 61.3%
Hence, the rate of return is 61.3%.
Calculate the average density in kilograms [kg] per cubic meter [m3] for a White Dwarf with a mass of
1 solar mass and the size of planet Earth. Be sure to use the correct units. (b) Calculate your own weight on the surface of the white dwarf star.
Answer:
(a) 2 × 10^9 kg/m^3; (b) roughly the mass of the Statue of Liberty.
Step-by-step explanation:
(a) Density of white dwarf:
D = m/V
Data:
1 solar mass = 2 × 10^30 kg
1 Earth radius = 6.371 × 10^6 m
Calculations:
V = (4/3)πr^3 = (4/3)π × (6.371 × 10^6 m)^3 = 1.083 × 10^21 m^3
D = 2 × 10^30 kg/1.083 × 10^21 m^3 = 2 × 10^9 kg/m^3
2. Weight on a white dwarf
The formula for weight is
w = kMm/r^2
where
k = a proportionality constant
M = mass of planet
m = your mass
w(on dwarf)/w(on Earth) = [kM(dwarf)m/r^2] /[kM(Earth)m/r^2
k, m, and r are the same on both planets, so
w(on dwarf)/w(on Earth) = M(dwarf)/M(Earth)
w(on dwarf) = w(on Earth) × [M(dwarf)/M(Earth)]
Data:
M(Earth) = 6.0 × 10^24 kg
Calculation:
w(on dwarf) = w(on Earth) × (2 × 10^30 kg /6.0 × 10^24 kg)
= 3.3 × 10^5 × w(on Earth)
Thus, if your weight on Earth is 60 kg, your weight on the white dwarf will be
3.3 × 10^5 × 60 kg = 2 × 10^7 kg
That's roughly as heavy as the Statue of Liberty is on Earth.
Find an equation for the line that passes through the points (-2, 1) and (6, -5)
Answer:
y=-3/4x-1/2
Step-by-step explanation:
Using the slope equation,
m=y₂-y₁
____
x₂-x₁
you get the slope:
m=-5-1/6+2
m=-6/8
m=-3/4
We find that the intercept is -1/2.
The equation is y=-3/4x-1/2
The answer is:
The equation of the line that passes through the points (-2,1) and (6,-5) is:
[tex]y=-\frac{3}{4}x-\frac{1}{2}[/tex]
Why?To solve the problem, we can use the following formula:
We have that:
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}*(x-x_1)[/tex]
So, using the given points (-2,1) and (6,-5), we have:
[tex]y-1=\frac{-5-1}{6-(-2)}*(x-(-2))[/tex]
[tex]y-1=\frac{-6}{6+2}*(x+2)[/tex]
[tex]y-1=\frac{-6}{8}*(x+2)[/tex]
[tex]y-1=-\frac{3}{4}*(x+2)[/tex]
[tex]y=-\frac{3}{4}*(x+2)+1[/tex]
[tex]y=-\frac{3}{4}*(x)-\frac{3}{4}**(2)+1[/tex]
[tex]y=-\frac{3}{4}(x)-\frac{6}{4})+1[/tex]
[tex]y=-\frac{3}{4}(x)-\frac{3}{2})+1[/tex]
[tex]y=-\frac{3}{4}x-\frac{1}{2}[/tex]
Hence, we have that the equation of the line that passes through the points (-2,1) and (6,-5) is:
[tex]y=-\frac{3}{4}x-\frac{1}{2}[/tex]
Have a nice day!
On a certain exam, Tony corrected 2020 papers and found the mean for his group to be 6060. Alice corrected the remaining 1010 papers and found that the mean for her group was 8080. What is the mean of the combined group of 3030 students
Answer:
The mean is 66.667 ( approx )
Step-by-step explanation:
Let x be the sum of Tony's group and y be the sum of Alice's group,
We know that,
[tex]Mean = \frac{\text{Total sum of observations}}{\text{Number of observations}}[/tex]
According to the question,
In Tony's group,
Students = 20,
Mean = 60,
[tex]\implies \frac{x}{20}=60\implies x = 1200[/tex]
In Alice's group,
Students = 10,
Mean = 80,
[tex]\implies \frac{y}{10}=80\implies y = 800[/tex]
Thus, the total sum of combined group of 30 students = 1200 + 800 = 2000,
Hence, the mean of the combined group = [tex]\frac{2000}{30}[/tex]
[tex]\approx 66.667[/tex]
The probability that a randomly selected individual in a certain community has made an online purchase is 0.35 . Suppose that a sample of 12 people from the community is selected, what is the probability that at most 3 of them has made an online purchase?
Answer:
The required probability is approximately 0.3467.
Step-by-step explanation:
Let X represents the event of making an online purchase,
Given,
The probability of making an online purchase, p = 0.35,
While, the probability of not making the online purchase, q = 1 - p = 0.65,
Hence, by the binomial distribution formula,
[tex]P(x) = ^nC_x p^x q^{n-x}[/tex]
Where, [tex]^nC_x=\frac{n!}{x!(n-x)!}[/tex]
Hence, the probability that at most 3 of them has made an online purchase is,
P(x ≤ 3) =P(x=0) + P(X=1) + P(X=2) + P(x=3)
[tex]= ^{12}C_0 p^0 q^{12-0}+^{12}C_1 p^1 q^{12-1}+^{12}C_2 p^2 q^{12-2}+^{12}C_3 p^3 q^{12-3}[/tex]
[tex]=(0.65)^{12}+12(0.35)(0.65)^{11}+66(0.35)^2(0.65)^{10}+220(0.35)^3(0.65)^9[/tex]
[tex]=0.346652696179[/tex]
[tex]\approx 0.3467[/tex]
To find the probability that at most 3 people in a sample of 12 have made an online purchase, use the binomial probability formula.
Explanation:To find the probability that at most 3 people in a sample of 12 have made an online purchase, we can use the binomial probability formula. The formula is P(X ≤ k) = Σ{k=0}^{k} (nCk) * p^k * (1-p)^(n-k), where n is the sample size, k is the number of successes, p is the probability of success, and (nCk) is the combination.
In this case, n = 12, k ≤ 3, p = 0.35. So, the probability is:
P(X = 0) = (12C0) * (0.35)^0 * (0.65)^(12-0)P(X = 1) = (12C1) * (0.35)^1 * (0.65)^(12-1)P(X = 2) = (12C2) * (0.35)^2 * (0.65)^(12-2)P(X = 3) = (12C3) * (0.35)^3 * (0.65)^(12-3)Then, you can sum up these probabilities to find the total probability that at most 3 people have made an online purchase.
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You buy a family-size box of laundry detergent that contains 48 cups. If your washing machine calls for 1 and 1/5 cups per wash load, how many loads of wash can you do?
Answer:
40 loads
Step-by-step explanation:
To find how many loads of wash you can do you need to divide 48 by 1 1/5.
There are two ways you can divide this, the first way is converting 48 to a fraction and dividing them.
48/1 divided by 1 1/5
convert 1 1/5 to an improper fraction
48/1 divided 6/5
change the division to multiplication and find the reciprocal of the second fraction.
48/1*5/6 = 240/6
Simplify to 40/1 or 40
The second way is changing 1 1/5 to a decimal, so its 1.2
Then divide 48 by 1.2 and you get 40.
By dividing the total amount of detergent by the amount required per load, you can determine that a 48-cup family-size box of laundry detergent can do 40 loads of wash.
Explanation:To find out how many loads of wash can be done with a family-size box of laundry detergent, you need to divide the total amount of detergent, 48 cups, by the amount required per load, which is 1 and 1/5 cups.
Firstly, we need to convert the mixed fraction into an improper fraction. 1 and 1/5 = 5/5 + 1/5 = 6/5.
Then, we do the division: 48 ÷ (6/5) = 48 * (5/6) = 40. This operation is equivalent to multiplying by the reciprocal of the fraction.
So, with a 48-cup family-size box of laundry detergent, you could do 40 loads of wash, assuming each load requires 1 and 1/5 cups of detergent.
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A business firm produces and sells a particular product. Variable cost is P30 per unit. Selling price is P40 per unit.
Fixed cost is P60,000. Determine the following:
a. Profit when sales are 10,000 units
b. The break-even point quantity and revenue
c. Sales when profits are at P9,000
d. The amount by which fixed is cost will have to be decreased or increased, to allow the firm to break even at sales volume of 500 units. Variable cost and selling price per unit remain constant.
e. The volume of sales to cover the fixed cost
f. Suppose that the firm want to break-even at a lower number of units, assuming that Fixed cost and Variable cost remain constant, how is the selling price affected?
Answer:
a.The profit is 40000 when sales are 10000 units.
b.Break-even point quantity and revenue=6000
c.When profits are at P9,000, sales are 6900
d.Fixed cost must decrease
e.The volume of sales to cover the fixed cost is 1500 units
f.If the firm want to break-even at a lower number of units, then the price will rice
Step-by-step explanation:
a.Profit is the difference between sales and cost
Profit= price* sales -((Variable cost * sales) +Fixed cost)
Profit when sales are 10000 units must be
P=40*10000-((30*10000)+60000)
P=400000-(300000+60000)=400000-360000
Profit=40000
The profit is 40000 when sales are 10000 units.
b.The break-even point quantity and revenue is when profit=0
So, Profit= price* sales -((Variable cost * sales) +Fixed cost)
If profit is 0, then (Variable cost * sales) +Fixed cost =price* sales
30x +60000=40x
10x=60000
x=60000/10=6000
Break-even point quantity and revenue=6000
c. Profit= price* sales -((Variable cost * sales) +Fixed cost)
9000=40x -(30x +60000)=40x -30x -60000)
9000 +60000=40x-30x
69000=10x
x=6900 units
d. break even at sales volume of 500 units
(Variable cost * sales) +Fixed cost =price* sales
30*500+FC=40*500
1500+FC=2000
FC=2000-1500
FC=500 Fixed cost must decrease
e.The volume of sales to cover the fixed cost
To only cover fixed cost, sales have to be 60000
Fixed cost =price* sales
Sales=Fixed cost/price
Sales 60000/40=1500 units
f. If the firm want to break-even at a lower number of units, then the price will rice
Remember that break-even formula is
(Variable cost * sales) +Fixed cost =price* sales
Variable an fixed cost remain constant, if sales go down, then price must go up.
A simple random sample of 10 households, the number of TV's that each household had is as follows: 2 , 0 , 2 , 2 , 2 , 2 , 1 , 5 , 3 , 2 Assume that it is reasonable to believe that the population is approximately normal and the population standard deviation is 0.55 . What is the lower bound of the 95% confidence interval for the mean number of TV's?
Answer: 1.758 is the lower bound of the 95% confidence interval for the mean number of TV's.
Step-by-step explanation:
Given that,
n = 10
Number of TV each household have = {2 , 0 , 2 , 2 , 2 , 2 , 1 , 5 , 3 , 2}
Standard Deviation(SD) = 0.55
95% Confidence Interval, = 0.05
Follows normal distribution,
Mean = [tex]\bar{X} = \frac{2+0+2+2+2+2+1+5+3+2}{10}[/tex]
= [tex]\frac{21}{10}[/tex]
= 2.1
Therefore, 95% Confidence Interval are as follows:
[tex]\bar{X}\pm Z_{\frac{\alpha}{2}} \times \frac{\sigma}{\sqrt{n}}[/tex]
[tex]2.1\pm 1.96 \times \frac{\0.55}{\sqrt{10}}[/tex]
Hence,
Lower bound = 2.1- 1.96 × [tex]\frac{\0.55}{\sqrt{10}}[/tex]
= 2.1- 1.96 × 0.174
= 1.758
True or false. If a is any odd integer, then a^2 + a is even. Explain this.
Answer:
True.
Step-by-step explanation:
We can represent an odd number by 2n + 1 where n = 0, 1, 2, 3, 5 etc.
Substituting:
a^2 + a = (2n + 1)^2 + 2n + 1
= 4n^2 + 4n + 1 + 2n + 1
= 4n^2 + 6n + 2
= 2(2n^2 + 3n + 1)
which is even because any integer multiplied by an even number is even.
This is also true if we use a negative odd integer:
We have 4n^2 + 4n + 1 - 1 - 2n
= 4n^2 + 2n
= 2(2n^2 + n(.
The statement is true. For any odd integer 'a', the expression 'a² + a' will always be even. This is because when 'a' (in the form of 2n+1 where n is any integer) is squared and added to 'a', the result is a number that is divisible by two, hence an even number.
Explanation:Your statement is true. If a is any odd integer, then a² + a is indeed even. Here's why:
Any odd number can be expressed in the form 2n+1, where n is any integer. So, when you square this you get (2n+1)² = 4n² + 4n + 1, which simplifies to 2(2n² + 2n) + 1. This is an odd number.
Then, if you add a (which is 2n+1), you get 2(2n² + 2n) + 1 + 2n + 1, which simplifies to 2(2n² + 3n + 1). This is divisible by 2, which means it's an even number. Therefore, the expression a² + a represents an even number.
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Suppose that in a senior college class of 500 students itis found that 210 smoke, 258 drink alcoholic beverages, 216 eatbetween meals, 122 smoke and drink alcoholic beverages, 83 eatbetween meals and drink alcoholic beverages, 97 smoke and eatbetween meals, and 52 engage in all three of these bad healthpractices. If a member of this senior class is selected at random,find the probability that the studenta.) smokes but does not drink alcoholic beverages;b.) eats between meals and drinks alcoholic beverages but doesnot smoke;c.) neither smokes nor eats between meals.Please show all steps to solve, not in books listed
Answer:
a)Smoke but doesn't drink alcoholic beverages
P=0.176 or 17.6%
b)eats between meals and drinks alcoholic beverages but doesn't smoke
P=0.062 or 6.2%
c.) neither smokes nor eats between meals
P=0.342 or 34.2%
Step-by-step explanation:
Using a Venn diagram with three circles, one for each bad health habit. Let us [tex]a[/tex] for the students that only smoke, [tex]b[/tex] for only drink alcoholic beverages, and [tex]c[/tex] for only eat between meals. Following this logic, [tex]d[/tex] represents smoke and drink alcoholic beverages but not eat between meals, [tex]e[/tex] drink alcoholic beverages and eat between meals but not smoke,[tex]f[/tex] eat between meals and smoke but not drink alcoholic beverages. Finally [tex]g[/tex] for having all the three, this means [tex]g=52[/tex].
To find [tex]f[/tex], subtract 52 (the three problems) from 97 because this last number represents all the students that smoke and eat between meals, including the students that have the three bad habits. The same goes for 'd' and 'e'.
[tex]f=97-52=45[/tex]
[tex]e=83-52=31[/tex]
[tex]d=122-52=70[/tex]
To find [tex]a[/tex], subtract [tex]e[/tex], [tex]f[/tex], and [tex]g[/tex] from the total of smokers. This is because [tex]e[/tex], [tex]f[/tex], and [tex]g[/tex] represent smoke and at least another bad habit and [tex]a[/tex] represents only smoking.
[tex]a=210-d-f-g=210-70-45-52=43[/tex]
The same goes for [tex]b[/tex] and [tex]c[/tex].
[tex]b=258-d-e-g=258-70-31-52=105[/tex]
[tex]c=216-e-f.-g=216-31-45-52=88[/tex]
Adding all letters and subtract from the total to see if there is any healthy student:
[tex]500-a+b+c+d+e+f+g =500-43+105+88+70+31+45+52=500- 434=66[/tex]
a)Smoke but doesn't drink alcoholic beverages
This will be 'a' (only smokes) and 'f' ( smokes and eats between meals but doesn't drink) divided by the total of students.
[tex]P=(43+45)/500=0.176[/tex]
b)eats between meals and drinks alcoholic beverages but doesn't smoke
This probability is 'e' divided by the total of students.
[tex]P=31/500=0.062[/tex]
c.) neither smokes nor eats between meals
This will be 'b' (only drinks) plus the healthy students (66) divided by the total of students.
[tex]P=(105+66)/500=0.342[/tex]
The probability a randomly selected student smokes but does not drink is 0.176, eats between meals and drinks alcohol but does not smoke is 0.062, and neither smokes nor eats between meals is 0.286.
Explanation:To solve this, we need to break down the information into three categories: those who smoke (S), those who drink alcoholic beverages (D), and those who eat between meals (E). We’re told that there are 210 smokers, 258 drinkers, and 216 who eat between meals. We also know that some students fall into multiple categories.
a) Find those who smoke but do not drink. We know that 122 students both smoke and drink, so the number who smoke but do not drink is 210 (total smokers) - 122 (smokers and drinkers) = 88. The probability is then 88 out of 500, or 0.176.
b) To find those who eat between meals and drink alcoholic beverages but do not smoke, we subtract those who do all three (52) from those who eat between meals and drink (83) to get 31. The probability is then 31 out of 500, or 0.062.
c) To find those who neither smoke nor eat between meals, we subtract those who do either from the total. We add together the numbers who smoke, drink, or eat between meals (210+258+216), then subtract off twice the number who do two activities (122+83+97) since they were counted twice in the first total, then add back in the number who do all three (52) since they were subtracted too many times. Subtracting this from 500 gives us the number who do neither. So, 500 - ((210+258+216)-(2*(122+83+97))+52) = 143. The probability is then 0.286.
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Consider the differential equation below. (You do not need to solve this differential equation to answer this question.) y' = y^2(y + 4)^3 Find the steady states and classify each as stable, semi-stable, or unstable. Draw a plot showing some typical solutions. If y(0) = -2 what happens to the solution as time goes to infinity?
We have [tex]y'=0[/tex] when [tex]y=0[/tex] or [tex]y=-4[/tex], so we need to check the sign of [tex]y'[/tex] on 3 intervals:
Suppose [tex]-\infty<y<-4[/tex]. In particular, let [tex]y=-5[/tex]. Then [tex]y'=(-5)^2(-5+4)^3=-25<0[/tex]. Since [tex]y'[/tex] is negative on this interval, we have [tex]y(t)\to-\infty[/tex] as [tex]t\to\infty[/tex].Suppose [tex]-4<y<0[/tex], say [tex]y=-1[/tex]. Then [tex]y'=(-1)^2(-1+4)^3=-27<0[/tex], so that [tex]y(t)\to-4[/tex] as [tex]t\to\infty[/tex].Suppose [tex]0<y<\infty[/tex], say [tex]y=1[/tex]. Then [tex]y'=1^2(1+4)^3=125>0[/tex], so that [tex]y(t)\to\infty[/tex] as [tex]t\to\infty[/tex].We can summarize this behavior as in the attached plot. The arrows on the [tex]y[/tex]-axis indicate the direction of the solution as [tex]t\to\infty[/tex]. We then classify the solutions as follows.
[tex]y=0[/tex] is an unstable solution because on either side of [tex]y=0[/tex], [tex]y(t)[/tex] does not converge to the same value from both sides.[tex]y=-4[/tex] is a semi-stable solution because for [tex]y>-4[/tex], solutions tend toward the line [tex]y=-4[/tex], while for [tex]y<-4[/tex] solutions diverge to negative infinity.An electronic product takes an average of 8 hours to move through an assembly line. If the standard deviation of 0.4 hours, what is the probability that an item will take between 8.4 and 9.1 hours to move through the assembly line?
Answer: 0.1557
Step-by-step explanation:
Given : Mean : [tex]\mu=\ 8[/tex]
Standard deviation : [tex]\sigma= 0.4[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
Let the random variable x (number of hours) is normally distributed .
For x= 8.4
[tex]z=\dfrac{8.4-8}{0.4}=1[/tex]
For x= 9.1
[tex]z=\dfrac{9.1-8}{0.4}=2.75[/tex]
The p-value =[tex] P(8.4<x<9.1)=P(1<z<2.75)[/tex]
[tex]=P(z<2.75)-P(z<1)= 0.9970202-0.8413447\\\\=0.1556755\approx0.1557[/tex]
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 7x6, y = 7x, x ≥ 0; about the x-axis
Answer:
The volume is [tex]\frac{490\pi}{39}[/tex] cubic units.
Step-by-step explanation:
The given curve is
[tex]y=7x^6[/tex]
The given line is
[tex]y=7x[/tex]
Equate both the functions to find the intersection point of both line and curve.
[tex]7x^6=7x[/tex]
[tex]7x^6-7x=0[/tex]
[tex]7x^6-7x=0[/tex]
[tex]7x(x^5-1)=0[/tex]
[tex]7x=0\rightarrow x=0[/tex]
[tex]x^5-1=0\rightarrow x=1[/tex]
According to washer method:
[tex]V=\pi \int_{a}^{b}[f(x)^2-g(x)^2]dx[/tex]
Using washer method, where a=0 and b=1, we get
[tex]V=\pi \int_{0}^{1}[(7x)^2-(7x^6)^2]dx[/tex]
[tex]V=\pi \int_{0}^{1}[49x^2-49x^{12}]dx[/tex]
[tex]V=49\pi \int_{0}^{1}[x^2-x^{12}]dx[/tex]
[tex]V=49\pi [\frac{x^3}{3}-\frac{x^{13}}{13}]_0^1[/tex]
[tex]V=49\pi [\frac{1^3}{3}-\frac{1^{13}}{13}-(0-0)][/tex]
[tex]V=49\pi [\frac{1}{3}-\frac{1}{13}][/tex]
[tex]V=49\pi (\frac{13-3}{39})[/tex]
[tex]V=49\pi (\frac{10}{39})[/tex]
[tex]V=\frac{490\pi}{39}[/tex]
Therefore the volume is [tex]\frac{490\pi}{39}[/tex] cubic units.
The volume of the solid obtained by rotating the region bounded by the curves y = 7x6 and y = 7x, for x ≥ 0, around the x-axis is determined by setting up and evaluating a volume integral using the method of cylindrical shells. The intersection points of the curves, that serve as the limits of the integral, are found by equating the two functions, giving x = 0 and x = 1.
Explanation:To find the volume of the solid obtained by rotating the area between the curves y = 7x6 and y = 7x around the x-axis, we use the method of cylindrical shells.
Firstly, we find the intersection points of the two curves by setting them equal to each other: 7x6 = 7x, which gives us x = 0 and x = 1.
Next, we set the volume integral and evaluate it over the range [0,1]. The general formula is V = 2π ∫ from a to b [x * (f(x) - g(x)) dx]. Here, f(x) = 7x6 and g(x) = 7x, therefore,
V = 2π ∫ from 0 to 1 [x * (7x6 - 7x) dx] = 2π ∫ from 0 to 1 [7x7 - 7x2 dx],
which can be integrated term-by-term. The antiderivatives of x7 and x2 are (1/8)x8 and (1/3)x3, respectively. By implementing these with the Fundamental Theorem of Calculus, we then substitute the limits of integration, subtract, and simplify to reach the final volume of the rotated solid.
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Suppose you have the formula =$D$5*E5 in cell F5. When you copy the formula into cell F6 what will the new formula be?
a. =$D$5*E5
b. =D5*E6
c. =$D$5*E6
d. =$D$6*E6
Answer:
C. =$D$5*E6
Step-by-step explanation:
In excel Columns are A, B, C, D, .....
And, rows are, 1, 2, 3, ....
Also, the intersection of column and row is called cell,
eg : A1 is a cell,
When we copy a function formula from a cell A1 to B2,
Then in the formula we replace A1 by B2 or vice versa,
But the row or column which comes after the dollar sign is anchored or absolute.
That is, when we copy an excel formula with $ sign they will copy cells referred in that formula relative to the position where they are being copied to.
Thus, if we have the formula =$D$5*E5 in cell F5,
Then, $D$5*E6 must be the new formula into cell 6.
Option 'C' is correct.
Select all the levels of measurement for which data can be qualitative.A.Nominal.B.IntervalC.RatioD.Ordinal
Select all the levels of measurement for which data can be qualitative.A.Nominal.B.IntervalC.RatioD.Ordinal
ratio
A patient needs atropine 0.2 mg IM STAT. It's available in 0.4 mg/mL. What's the correct dosage for this patient?
A. 0.5 mL B. 0.1 mL C. 1 mL D. 5 mL
Answer:
The correct dosage is 0.5 ml
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.
We can solve this question by using the simple Rule of Three property. The Property is the following
[tex]\frac{a}{x} = \frac{b}{c}[/tex] ⇒ [tex]x = \frac{a*c}{b}[/tex]
Now we can use the property above using the values given to us to find the correct dosage for the patient.
[tex]\frac{0.2mg}{x} = \frac{0.4mg}{1ml}[/tex]
[tex]x = \frac{0.2mg*1ml}{0.4mg}[/tex]
[tex]x = \frac{0.2ml}{0.4}[/tex]
[tex]x = 0.5ml[/tex]
So Now we can see that the correct dosage is 0.5 ml
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
g Use the counting principle to determine the number of elements in the sample space. The possible ways to complete a multiple-choice test consisting of 20 questions, with each question having four possible answers (a, b, c, or d).
Answer:
[tex](4)^{20}[/tex]
Step-by-step explanation:
Total number of questions = 20
Possible options for each question = 4
Sample space contains the total number of possible outcomes.
For every question there are 4 possible ways to select an answer. This holds true for all 20 questions. Selecting an answer for a question is independent of other questions/answers,
According to the counting principle, the total number of possible outcomes will be the product of the number of possible outcomes of individual events. Possible outcomes for each of the 20 questions is 4. This means we have to multiply 4 twenty times to find the total number of possible outcomes.
So, the number of elements in the sample space would be:
[tex](4)^{20}[/tex]
18. Polar Bear Frozen Foods manufactures frozen French fries for sale to grocery store chains. The final package weight is thought to be a uniformly distributed random variable. Assume X, the weight of French fries, has a uniform distribution between 50 ounces and 68 ounces. What is the mean weight for a package? What is the standard deviation for the weight of a package? Round your answers to four decimal places, if necessary.
Answer:
μ = 59, σ = 5.1962
Step-by-step explanation:
For a uniform distribution, where a is the minimum and b is the maximum, the mean (or average) is:
μ = (a + b) / 2
And the standard deviation is:
σ = (b − a) / √12
Here, a = 50 and b = 68.
The mean is:
μ = (50 + 68) / 2
μ = 59
And the standard deviation is:
σ = (68 − 50) / √12
σ = 5.1962
The mean weight of a package of French fries produced by Polar Bear Frozen Foods is 59 ounces. The standard deviation of the weight is approximately 5.1962 ounces.
Explanation:The question pertains to understanding the mean and standard deviation of a uniformly distributed variable, specifically the weight of French fries produced by Polar Bear Frozen Foods. A uniform distribution is a type of probability distribution that has constant probability.
The formula for the mean (or average) of a uniform distribution is (a + b) / 2 where 'a' is the minimum value and 'b' is the maximum value. Given in the question a = 50 ounces and b = 68 ounces, we obtain the mean as (50+68)/2 = 59 ounces.
The formula for the standard deviation of a uniform distribution is sqrt[(b-a)^2 /12 ]. Thus, substituting values we get sqrt[(68-50)^2 / 12] = sqrt[324/12] = sqrt[27] = 5.1962 ounces (rounded to four decimal places).
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A cellphone service provider charges $5.00 per month and $0.20 per minute per call. If a customer's current bill is $55, how many minutes did the customer use? (Round any intermediate calculations and your final answer to the nearest whole minute.) 300 minutes 275 minutes 250 minutes 225 minutes
Answer:
250 minutes
Step-by-step explanation:
Given,
Cell phone charges for a month = $ 5.00,
Additional charges per minute = $0.20,
Let the cell phone is used for x minutes for a month such that the total bill is $ 55,
⇒ Cell phone charge for the month + additional charges for x minutes = $ 55
⇒ 5.00 + 0.20x = 55
Subtracting 5 on both sides,
0.20x = 50
x = 250,
Hence, the cell phone is used for 250 minutes.
Third option is correct.
To offer scholarships to children ofâ employees, a company invests 10,000 at the end of every three months in an annuity that pays 8.5% compounded quarterly.
a. How much will the company have in scholarship funds at the end of tenâ years?
b. Find the interest.
a. The company will have $... in scholarship funds.
Answer:
a. $633 849.78; b. $233 849.78
Step-by-step explanation:
a. Value of Investment
The formula for the future value (FV) of an investment with periodic deposits (p) is
FV =(p/i)(1 + i)[(1 + i)^n -1)/i]
where
i = interest rate per period
n = number of periods
Data:
p = $10 000
APR = 8.5 % = 0.085
t = 10 yr
Calculations:
Deposits are made every quarter, so
i = 0.085/4 = 0.02125
There are four quarters per year, so
n = 10 × 4 = 40
FV = (10 000/0.02125)(1 + 0.02125)[(1 + 0.02125)^40 - 1)]
= 470 588.235 × 1.02125 × (1.02125^40 - 1)
= 480 588.235(2.318 904 06 - 1)
= 480 588.235 × 1.318 904 06
= 633 849.78
The company will have $633 849.78 in scholarship funds.
b. Interest
Amount accrued = $633 849.78
Amount invested = 40 payments × ($10 000/1 payment) = 400 000.00
Interest = $233 849.78
The scholarship fund earned $233 849.78 in interest.
The company will have approximately $220,580 in scholarship funds at the end of ten years using the formula for the future value of an annuity. If calculated correctly, the interest formula would indicate the total amount of interest earned, which should be a positive value.
Explanation:To calculate how much the company will have in scholarship funds at the end of ten years, we use the future value formula of an annuity. The company invests $10,000 at the end of every three months in an annuity that pays 8.5% interest compounded quarterly. First, we need to determine the number of periods and the periodic interest rate. Since the investments are made quarterly, there are 4 periods in a year. Over ten years, there are 4 * 10 = 40 periods. The periodic interest rate is 8.5% per year, or 8.5%/4 = 2.125% per period.
Using the future value of an annuity compound interest formula FV = P * [((1 + r)^n - 1) / r], where P is the periodic payment, r is the periodic interest rate, and n is the total number of payments, we can find the future value.
In this case, P = $10,000, r = 2.125% (or 0.02125 as a decimal), and n = 40. Plugging these values into the formula, we get:
FV = $10,000 * [((1 + 0.02125)^40 - 1) / 0.02125]
FV = $10,000 * [(1.02125^40 - 1) / 0.02125]
FV = $10,000 * [2.2058...]
FV = $220,580...
Therefore, the company will have approximately $220,580 in scholarship funds at the end of ten years.
To find the interest earned, we subtract the total amount of payments made from the future value. The total amount of payments is $10,000 * 40 = $400,000. So the interest earned is $220,580 - $400,000 = $-179,420. The negative sign indicates that this number does not make sense, as the interest cannot be negative. This is an error, and we should re-calculate:
Total investments = $10,000 * 40 = $400,000
Interest = Future Value - Total Investments
Interest = $220,580 - $400,000 = $-179,420 (This is incorrect)
To correct this, we should correctly apply the future value formula once more and make sure all calculations are done precisely. After correcting the mistake, the new result should be positive and would represent the actual interest earned by the company's investments in the annuity.
A fair die is rolled fourfour times. A 2 is considered "success," while all other outcomes are "failures." Find the probability of 4 successessuccesses.
Hence, the probability is:
[tex]\dfrac{1}{6^4}\ or\ 0.000772[/tex]
Step-by-step explanation:It is given that:
A fair die is rolled four times. A 2 is considered "success," while all other outcomes are "failures."
This means that the probability of 4 successes is the outcome such that each of the four die will result in the outcome 2.
Also, the probability of 2 in each of the die is: 1/6
( since, there are total 6 outcomes in a die {1,2,3,4,5,6} and out of which there is only one '2' )
Also, we know that the outcome on one die is independent on the other this means that the probability of 4 successes is:
[tex]=\dfrac{1}{6}\times \dfrac{1}{6}\times \dfrac{1}{6}\times \dfrac{1}{6}\\\\\\\\=\dfrac{1}{6^4}\\\\\\=0.000772[/tex]
Final answer:
The probability of rolling a 2 four times in a row on a fair six-sided die is found by multiplying the probability of a single 2 (which is 1/6) four times, giving us a final probability of 1/1296 or approximately 0.0008.
Explanation:
To find the probability of rolling a 2 on a six-sided die four times in a row, we consider each roll as an independent event. The probability of rolling a 2 on each individual roll is 1/6, since there are six faces on the die and only one face with a 2 on it.
Since these events are independent, the joint probability of all four events occurring is the product of the individual probabilities:
Probability of 4 successes (rolling a 2 four times) = (1/6) * (1/6) * (1/6) * (1/6) = 1/1296.
This is computed by multiplying the probability of a single success, 1/6, four times since the dice rolls are independent events. Therefore, the probability of obtaining four successes is quite low at approximately 0.0008 when rounded to four decimal places.
How many 2 card hands are possible with a 52-card deck?
Answer:
2,652
Step-by-step explanation:
51*52=2652
(3 points + 1 point BONuS) Many people grab a granola bar for breakfast or for a snack to make it through the afternoon slump at work. A Kashi GoLean Crisp Chocolate Caramel bar weights 45 grams. The mean amount of protein in each bar is 7.8 grams. Suppose the distribution of protein in a bar is normally distributed with a standard deviation of 0.2 grams and a random Kashi bar is selected. (0.5 pts.) a) What is the probability that the amount of protein is between 7.65 and 8.2 grams?
Answer: 0.7506
Step-by-step explanation:
Given :Mean : [tex]\mu=\text{ 7.8 grams}[/tex]
Standard deviation : [tex]\sigma =\text{ 0.2 grams}[/tex]
The formula for z -score :
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 7.65 ,
[tex]z=\dfrac{7.65-7.8}{0.2}=-0.75[/tex]
For x= 8.2 ,
[tex]z=\dfrac{8.2-7.8}{0.2}=2[/tex]
The p-value = [tex]P(-0.75<z<2)=P(z<2)-P(z<-0.75)[/tex]
[tex]=0.9772498-0.2266274=0.750622\approx0.7506[/tex]
Hence, the probability that the amount of protein is between 7.65 and 8.2 grams=0.7506.
Final answer:
To calculate the probability of a Kashi GoLean Crisp Chocolate Caramel bar having a protein content between 7.65 and 8.2 grams, we use the Z-score formula. The probability is found to be approximately 75.06%.
Explanation:
To find the probability that a randomly selected Kashi GoLean Crisp Chocolate Caramel bar has a protein content between 7.65 and 8.2 grams, we use the Z-score formula for a normal distribution, where Z = (X - μ) / σ. Here, μ (mu) is the mean, and σ (sigma) is the standard deviation. The mean (μ) is 7.8 grams, and the standard deviation (σ) is 0.2 grams.
To find the Z-scores for 7.65 grams and 8.2 grams:
Z for 7.65 = (7.65 - 7.8) / 0.2 = -0.75Z for 8.2 = (8.2 - 7.8) / 0.2 = 2Next, we consult the standard normal distribution table to find the probabilities corresponding to these Z-scores. For Z = -0.75, the probability is approximately 0.2266, and for Z = 2, it's approximately 0.9772.
The probability that the protein is between 7.65 and 8.2 grams is the difference between these two probabilities: 0.9772 - 0.2266 = 0.7506 or 75.06%.
Use Laplace transforms to solve the initial value problem, then give the value of x(?).
x'' + 4x = 0; x(0) = 5, x'(0) = 0
Answer:
x = 5 cos 2t
Step-by-step explanation:
given equation
x'' + 4x = 0 ; x(0) = 5, x'(0) = 0
L{ x'' + 4 x } = 0
L{x''} + 4 L{x} = 0
s² . L(x) - s . x(0) - x'(0) + 4 L{x} = 0
( s² + 4 ). L(x) - 5 s = 0
L(x) = [tex]\dfrac{5s}{s^2 +4}[/tex]
[tex]L(\dfrac{s}{s^2 +a^2})[/tex] = cos at
so,
x = 5 [tex]L^{-1}(\dfrac{s}{s^2 +2^2})[/tex]
x = 5 cos 2t
Which of the following is equal to 6^5 - 6^4 / 5 = ?
a) 1/5 b) 6/5 c) 6^3 d) 6^4 / 5 e) 6^4
Answer:
[tex]6^4[/tex]
Step-by-step explanation:
We'd solve the exponents first:
[tex]6^5 = 7776[/tex]
[tex]6^4 = 1296[/tex]
Subtract:
7776 - 1296 = 6480
Divide:
[tex]6480\div5 = 1296[/tex]
We already know [tex]6^4 = 1296[/tex]
Our answer is [tex]6^4[/tex]
Answer:
6^4
Step-by-step explanation:
6^5 - 6^4 / 5 = ?
Factor out a 6^4
6^4(6-1)
---------------
5
Simplify
6^4(5)
---------------
5
Cancel the 5's
6^4