PLEASE HELP

Write equations for the vertical and horizontal lines passing through the point (-6, -5)

Answer:

• (x, y, z) = (3+t, 1-t, 8+4t) . . . equation of the line

• xy-intercept (1, 3, 0)

• yz-intercept (0, 4, -4)

• xz-intercept (4, 0, 12)

Step-by-step explanation:

The line's direction vector is given by the coordinates of the plane: (1, -1, 4). So, the parametric equations can be ...

(x, y, z) = (3, 1, 8) + t(1, -1, 4) . . . . . parametric equation for the line

or

(x, y, z) = (3+t, 1-t, 8+4t)

__

The various intercepts can be found by setting the respective variables to zero:

xy-plane: z=0, so t=-2. (x, y, z) = (1, 3, 0)

yz-plane: x=0, so t=-3. (x, y, z) = (0, 4, -4)

xz-plane: y=0, so t=1. (x, y, z) = (4, 0, 12)

This means that there exist constant a,b such that if:

                                au+bv=0

                             then a=b=0

{u+v,u-v} is a linearly independent set.

Let us consider there exists constant c,d such that:

                            c(u+v)+d(u-v)=0

To show:   c=d=0

The expression could also be written as:

 cu+cv+du-dv=0

( Since, using the distributive property)

Now on combining the like terms that is the terms with same vectors.

cu+du+cv-dv=0

i.e.

(c+d)u+(c-d)v=0

Since, we are given that u and v are linearly independent vectors this means that:

c+d=0------------(1)

and c-d=0 i.e c=d-----------(2)

and from equation (1) using equation (2) we have:

2c=0

i.e. c=0

and similarly by equation (2) we have:

         d=0

Hence, we are proved with the result.

We get that the vectors {u+v,u-v} is linearly independent.

The diver's current depth can be represented using signed numbers by subtracting the upward distance swum from the initial depth.

To represent the diver's current depth, we need to subtract the distance the diver has swum upward from the initial depth. The diver starts at -480 feet below the surface and swims upward 248 feet. Using signed numbers, we can represent the diver's current depth as -480 + 248 = -232 feet below the surface.

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Answer:

3200 people

Step-by-step explanation:

p = The frequency of the dominant gene

q = The frequency of the recessive gene

[tex]q^2=\frac{400}{10000}\\\Rightarrow q^2=0.04\\\Rightarrow q=0.2[/tex]

p+q = 1

⇒p = 1-q

⇒p = 1-0.2

⇒p = 0.8

Hardy-Weinberg formula

p² + 2pq + q² = 1

Now for heterozygous trait

2pq = 2×0.8×0.2 = 0.32

Multiplying with the population

0.32×10000 = 3200

∴ 3200 people would be expected to be heterozygous for this trait.

According to the Hardy-Weinberg formula, the expected number of people heterozygous for the eye color trait can be calculated as 768 in a population of 10,000. This calculation takes into account the dominance of the brown eye color trait and the frequency of blue-eyed individuals.

In this scenario, we are considering a single gene controlling the trait for eye color, with brown eyes being completely dominant to blue eyes.

Using the Hardy-Weinberg formula, we can calculate the expected frequency of each genotype in the population. The formula is: p^2 + 2pq + q^2 = 1.

We are given that 400 people have blue eyes in a population of 10,000. Therefore, the frequency of the recessive allele (q) can be calculated as the square root of the frequency of the blue-eyed individuals, which is 400/10,000 = 0.04.

Since brown eyes are completely dominant, the frequency of the dominant allele (p) can be calculated as 1 - q, which is 1 - 0.04 = 0.96.

Now we can calculate the expected number of heterozygous individuals (2pq): 2 * 0.96 * 0.04 * 10,000 = 768.

Therefore, we would expect 768 people to be heterozygous for the eye color trait in this population.

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Answer: (0.541, 0.819)

Step-by-step explanation:

The confidence interval for proportion is given by :-

[tex]p\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}[/tex]

Given : The proportion of children attended the school = [tex]p=\dfrac{51}{75}=0.68[/tex]

Significance level : [tex]\alpha=1-0.99=0.01[/tex]

Critical value : [tex]z_{\alpha/2}=z_{0.005}=\pm2.576[/tex]

Now, the 99% z ‑confidence interval for proportion will be :-

[tex]0.68\pm (2.576)\sqrt{\dfrac{0.68(1-0.68)}{75}}\approx0.68\pm 0.139\\\\=(0.68-0.139,0.68+0.139)=(0.541,\ 0.819)[/tex]

Hence, the 99% z ‑confidence interval for p, the proportion of all children enrolled in kindergarten who attended preschool = (0.541, 0.819)

Answer:

75.8%

Step-by-step explanation:

Mean weight of chickens = u = 1700 g

Standard deviation = [tex]\sigma[/tex] = 200g

We need to calculate the percentage of chickens having weight more than 1560 g

So,

x = 1560 g

Since the weights can be approximated by normal distribution, we can use concept of z-score to solve this problem.

First we need to convert the given weight to z score. The formula for z score is:

[tex]z=\frac{x-u}{\sigma}[/tex]

Using the values, we get:

[tex]z=\frac{1560-1700}{200} \\\\ z = -0.7[/tex]

So now we have to calculate what percentage of values lie above the z score of -0.7. Using the z-table or z-calculator we get:

P(z > -0.7) = 0.758

This means 0.758 or 75.8% of the values are above z score of -0.7. In context of our question we can write:

75.8% of the chickens will have weight more than 1560 g

To find the percent of chickens having weights more than 1560 g, calculate the z-score for 1560 g and find the area to the right of this z-score in the standard normal distribution curve.

To find the percent of all chickens having weights more than 1560 g, we need to calculate the z-score for 1560 g and then find the area to the right of this z-score in the standard normal distribution curve.

First, calculate the z-score using the formula: z = (x - μ) / σ, where x is the weight of the chicken, μ is the mean weight, and σ is the standard deviation.

For the weight 1560 g, the z-score is calculated as: z = (1560 - 1700) / 200 = -0.7

Using a standard normal distribution table or calculator, find the area to the right of -0.7. This area represents the percent of chickens having weights more than 1560 g.

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Answer:

The correct option is C. Fredo

Step-by-step explanation:

In a Mario Puzo's fictional novel named The Godfather, Frederico Corleone or Fredo is a fictional character. In the novel, Fredo's father is killed by the assasins. Witnissing his father being shot, Fredo goes into a shock.

To protect and aid Fredo's recovery, his elder brother Sonny, sends him to Las Vegas.

Therefore, Fredo is sent to Las Vegas

Answer:

[tex]P(w)=2w+\frac{2}{w}[/tex]

Step-by-step explanation:

We are given the area of a rectangle is 1 inch square.

You can find the area of a rectangle if you know the dimensions. Let's pretend the dimensions are w and l.

So we given w*l=1.

Now the perimeter of a rectangle with dimensions l and w is 2w+2l.

We want to express P=2w+2l in terms of w only.

We are given that w*l=1 so l=1/w (just divided both sides of w*l=1 by w).

So let's plug it in for l (the 1/w thing).

[tex]P=2w+2(\frac{1}{w})[/tex]

So [tex]P(w)=2w+\frac{2}{w}[/tex].

Answer:

P (w) = [tex]\frac{2}{w} +2w[/tex]

Step-by-step explanation:

We are given that the area of a rectangle is 1 square inches and we are to express the perimeter [tex]P(w)[/tex] as a function of the width [tex]w[/tex].

We know that:

Area of a rectangle = [tex]l \times w[/tex]

Substituting the given value of area in the above formula:

[tex]1=l \times w[/tex]

[tex]l=\frac{1}{w}[/tex]

Perimeter of a rectangle = [tex]2(l +w)[/tex]

Substituting the values in the formula to get:

Perimeter = [tex]2(\frac{1}{w}+w) =  \frac{2}{w} +2w[/tex]

Answer:

  29 months

Step-by-step explanation:

My TVM solver says that balance will be reached after about 29 monthly payments.

This question pertains to compound interest. You are depositing $270 monthly into an account with a monthly compound interest rate of 4.8%. By using the compound interest formula with logarithmic adjustments for monthly deposits, you can determine how long it will take you to save $8200.

The subject of the question is how long it would take to save up $8,200 for a boat by making $270 monthly deposits into an account that has a monthly compound interest rate of 4.8%. This is a question of compound interest. The formula for compound interest is A = P (1 + r/n)^(nt), where A is the total amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (in decimal form), n is the number of times that interest is compounded per year, and t is the time the money is invested for, in years.

In this case, we require to find 't' when we have A = $8200 , P = $270 (deposited every month), r = 4.8% (in decimal form, it becomes 0.048) and n = 12 (compounded monthly). However, as $270 is getting compounded every month, a slightly adjusted formula to calculate the number of months, t is required which is t = [log(A/P)] /[n * log(1 + r/n)]. By substituting A = $8200 and P = $270 and other values to this formula, we can find the time needed. This would require logarithmic math which is done usually in high school math courses or higher.

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Answer: Dashed Arrows

Step-by-step explanation:

Connector lines speak to arrangement streams when they interface two items in the equivalent BPMN pool. Items in various BPMN pools can't be associated by grouping stream, however they can synchronize through message stream. A connector line between two items in various pools that speaks to a message stream shows with a dashed line. Moving an article starting with one pool then onto the next likewise breaks the arrangement stream and changes over the association with a message-style line.

In a BPMN diagram, actors are represented by swimlanes, which denote responsibilities within a process and can be assigned to individuals, systems, or organizational units.

In a Business Process Model and Notation (BPMN) diagram, the actors are represented by swimlanes. These swimlanes are horizontal or vertical rectangles and they denote the different responsibilities within a process. Each swimlane is often dedicated to one actor, which can be a person, a system, or an organization unit involved in the process. For example, in a loan application process, there can be swimlanes representing the applicant, the loan officer, and the credit check system.

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Answer:

The standard form of required line is x-y=7.

Step-by-step explanation:

The standard form of a line is

[tex]ax+by=c[/tex]

Where, a,b,c are integers with no factor common to all three and a≥0.

The give equation of line is

[tex]x+y=2[/tex]

Here a=1 and b=1.

The slope of a standard line is

[tex]m=\frac{-a}{b}[/tex]

[tex]m_1=\frac{-1}{1}=-1[/tex]

The product of slops of two perpendicular lines is -1.

[tex]m_1\cdot m_2=-1[/tex]

[tex](-1)\cdot m_2=-1[/tex]

[tex]m_2=1[/tex]

The slope of required line is 1.

The point slope form of a line is

[tex]y-y_1=m(x-x_1)[/tex]

Where, m is slope.

The slope of required line is 1 and it passes through the point (1,-6). So, the equation of required line is

[tex]y-(-6)=1(x-1)[/tex]

[tex]y+6=x-1[/tex]

Add 1 on each side.

[tex]y+7=x[/tex]

Subtract y from both the sides.

[tex]7=x-y[/tex]

Therefore the standard form of required line is x-y=7.

Answer:

1. No

2. 7

3. b=54

Step-by-step explanation:

1. We can answer this by assuming a number.

Let our number be 23

Multiplying its digits = 6

Multiplying the result with initial number = 6 * 23 = 138

So it is not possible to get 1716 as a result by thinking of a natural number and applying the operation mentioned in the question.

2. What is the largest prime factor of the factorial 49! ?

First of all we have to define prime factors:

Prime factors are the prime numbers that can be multiplied together to equal the original number.

The factors of 49 are: 1, 7, 49

7 is the largest prime factor of 49

3. The GCD(a, b) = 18, LCM(a, b) = 108. If a=36, findb.

We will use the relationship:

[tex]GCD * LCM = a*b\\18*108=36b\\1944=36b\\b= \frac{1944}{36} \\b=54[/tex]

..

[tex]\bf cos^2(x)+2cos(x)+1=0\implies \stackrel{\textit{let's notice, this is simply }ax^2+bx+c=0}{[cos(x)]^2+2cos(x)+1=0} \\[2em] [cos(x)+1][cos(x)+1]=0 \\\\[-0.35em] ~\dotfill\\\\ cos(x)+1=0\implies cos(x)=-1\implies x=cos^{-1}(-1)\implies \stackrel{\textit{for the range }[0,2\pi ]}{x=\pi } \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \stackrel{\textit{all solutions}}{x=\pi +2\pi n~~,~~ n \in \mathbb{Z}}~\hfill[/tex]

Answer:

[tex]6ft[/tex] length on the east and west sides

[tex]12ft[/tex] length on the north and south sides

Step-by-step explanation:

Using x for the length of the east side (and is equal to the length  of the west side) and y for the length of the north side (and is equal to the length  of the south side), the equation that gives the total price equalized to 480 is:

[tex]20x+20x+10y+10y=480[/tex]

[tex]40x+20y=480[/tex]

Solving for y

[tex]y=\frac{-40x+480}{20}[/tex]

[tex]y=-2x+24[/tex]

The area of the garden is [tex]A=xy[/tex], to find the largest, substitute y in the formula of the area

[tex]A=x(-2x+24)=-2x^2+24x[/tex]

For the optimization, find the largest area, is needed the critical point. To find this point, derive A and equalize the derivative to zero:

[tex]A'=-4x+24=0[/tex]

Solve for x:

[tex]-4x=-24[/tex]

[tex]x=\frac{-24}{-4}[/tex]

[tex]x=6[/tex]

To see if x=6 is a maximum or a minimum, derive A' and substitute with x=6

[tex]A''=-4[/tex]

In this case, the second derivative of A doesn't depend on x, and it has a negative value, meaning the value found is a maximum. Using x=6 to find y

[tex]y=-2x+24[/tex]

[tex]y=-2(6)+24[/tex]

[tex]y=12[/tex]

The area is:

[tex]A=xy=6*12=72 ft^2[/tex]

For this case we must find the reversal of the following function:[tex]f (x) = 4x[/tex]

For it:

We change[tex]f (x)[/tex] by y:[tex]y = 4x[/tex]

We exchange the variables:

[tex]x = 4y[/tex]

We cleared "y":

[tex]y = \frac {x} {4}[/tex]

We change y for [tex]f^{-1}(x)[/tex]:

[tex]f ^ {- 1} (x) = \frac {x} {4}[/tex]

Answer:

The inverse of the given function is:[tex]f ^ {-1} (x) = \frac {x} {4}[/tex]

Step-by-step explanation:

The probability of getting tails each time is 1/2.  Seven times in a row, the probability is:

P = (1/2)^7

P = 0.0078125

the probability of obtaining seven tails in a row when flipping a coin is approximately 0.78%.

The probability of obtaining seven tails in a row when flipping a coin can be calculated using the principle of independent events in probability. Each flip of the coin is an independent event with two possible outcomes: heads or tails.

Probability can be calculated by using the formula:

[tex]P = \frac{number\ of\ desired\ outcomes}{total\ outcomes}[/tex]

here the desired outcome is 1 as we only need tails so we can say that:

[tex]P(tails) = \frac{1}{2} = 0.5[/tex]

To find the probability of obtaining seven tails in a row, you need to multiply the probability of getting tails on each individual flip:

[tex]Probability (7\ tails\ in\ a\ row) = 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5\\\\Probability (7\ tails\ in\ a\ row) = (0.5)^7 \approx 0.0078125[/tex]

Therefore, the probability of obtaining seven tails in a row when flipping a coin is approximately 0.78%.

Answer: Option(e) exist 123.50 is correct.

Step-by-step explanation:

James earns:

Product A: 15 × 35.75 = 536.25

Product B: 16 × 42.25 = 676

Total Earnings = 1212.25

Sally earns:

Product A: 25 × 35.75 = 893.75

Product B: 10 × 42.25 = 422.5

Total Earnings = 1316.25

Andre earns:

Product A: 18 × 35.75 = 643.5

Product B: 13 × 42.25 = 549.25

Total Earnings = 1192.75

Above calculation shows that Sally is the most profitable seller and Andre is the least profitable seller.

So, the difference between the most profitable seller i.e Sally (1316.25) and the least profitable seller i.e. Andre (1192.75) is 123.50.

Answer:

  1500 ft²

Step-by-step explanation:

The sum of two adjacent sides of the pasture is half the perimeter (160 ft/2 = 80 ft), so the side adjacent to the 50 ft side will be 80 ft - 50 ft = 30 ft.

The product of adjacent sides of a rectangle gives the area of the rectangle. That area will be ...

  area = (50 ft)(30 ft) = 1500 ft²

Answer: 0.2401

Step-by-step explanation:

The binomial distribution formula is given by :-

[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]

where P(x) is the probability of x successes out of n trials, p is the probability of success on a particular trial.

Given : The probability that a randomly chosen citizen-entity of Cygnus is of pension age† is approximately: p =0.7.

Number of trials  : n= 4

Now, the required probability will be :

[tex]P(x=4)=^4C_4(0.7)^4(1-0.7)^{4-4}\\\\=(1)(0.7)^4(1)=0.2401[/tex]

Thus, the probability that, in a randomly selected sample of four citizen-entities, all of them are of pension age =0.2401

Answer:

7

Step-by-step explanation:

[tex]x^2-y^2[/tex] is a difference of squares.

When factoring a difference of squares, you can use this formula [tex]u^2-v^2=(u-v)(u+v)[/tex].

So [tex]x^2-y^2[/tex] can be factored as [tex](x-y)(x+y)[/tex].

So back to the problem:

[tex]x^2-y^2=56[/tex]

Rewriting with a factored left hand side:

[tex](x-y)(x+y)=56[/tex]

We are given x-y=4 so rewriting again with this substitution:

[tex]4(x+y)=56[/tex]

Dividing both sides by 4:

[tex](x+y)=14[/tex]

So we have x+y equals 14.

We are asked to find the average of x and y which is (x+y)/2.

So since x+y=14 , then (x+y)/2=14/2=7.

Answer: 1.64

Step-by-step explanation:

80% = 50

20% = 12.5

100% = 62.5

38% = 1.64

Answer: The sum of five consecutive even numbers for this sequence is 20n.

Step-by-step explanation:

Since we have given that

Number of consecutive even numbers = 5

Middle value = 4n

Since there are 5 consecutive even numbers:

4n-4,4n-2,4n,4n+2,4n+4

So, Sum of five consecutive even numbers would be

[tex]4n-4+4n-2+4n+4n+2+4n+4\\\\=20n[/tex]

Hence, the sum of five consecutive even numbers for this sequence is 20n.

Answer:

He invest for 2 years.

Step-by-step explanation:

Given : Suppose you have $1,950 in your savings account at the end of a certain period of time. You invested $1,700 at a 6.88% simple annual interest rate.

To find : How long, in years, did you invest your money?

Solution :

Applying simple interest formula,

[tex]A=P(1+r)^t[/tex]

Where, A is the amount A=$1950

P is the principal P=$1700

r is the interest rate r=6.88%=0.0688

t is the time

Substitute the values in the formula,

[tex]1950=1700(1+0.0688)^t[/tex]

[tex]\frac{1950}{1700}=(1.0688)^t[/tex]

[tex]1.147=(1.0688)^t[/tex]

Taking log both side,

[tex]\log(1.147)=\log ((1.0688)^t)[/tex]

Applying logarithmic formula, [tex]\log a^x=x\log a[/tex]

[tex]\log(1.147)=t\log (1.0688)[/tex]

[tex]t=\frac{\log(1.147)}{\log (1.0688)}[/tex]

[tex]t=2.06[/tex]

Approximately, He invest for 2 years.

Answer:

1 machine must be made to minimise the unit cost.

Step-by-step explanation:

Step 1: Identify the function

x is the number of machines

C(x) is the function for unit cost

C (x) = 0.5x^2-150 + 21,035

Step 2: Substitute values in x to find the unit cost

C (x) = 0.5x^2-150 + 21,035

The lowest value of x could be 1

To check the lowest cost, substitute x=1 and x=2 in the equation.

When x=1

C (x) = 0.5x^2-150 + 21,035

C (x) = 0.5(1)^2-150 + 21,035

C (x) = 20885.5

When x=2

C (x) = 0.5x^2-150 + 21,035

C (x) = 0.5(2)^2-150 + 21,035

C (x) = 20887

We can see that when the value of x i.e. the number of machines increases, per unit cost increases.

Therefore, 1 machine must be made to minimise the unit cost.

!!

The unit cost is minimized when 150 machines are made.

To find the number of machines that must be made to minimize the unit cost, we need to find the minimum value of the function [tex]\( C(x) = 0.5x^2 - 150x + 21,035 \).[/tex] This can be done by finding the vertex of the quadratic function, as the vertex corresponds to the minimum (or maximum) value of the function.

The vertex of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] is given by the formula:

[tex]\[ x = \frac{-b}{2a} \][/tex]

Given the function [tex]\( C(x) = 0.5x^2 - 150x + 21,035 \)[/tex], we can see that [tex]\( a = 0.5 \) and \( b = -150 \).[/tex]

[tex]\[ x = \frac{-(-150)}{2 \cdot 0.5} \]\[ x = \frac{150}{1} \]\[ x = 150 \][/tex]

So, the number of machines that must be made to minimize the unit cost is 150.

Answer:  Probability that she attended class regularly given that she receives A grade is 0.9286.

Step-by-step explanation:

Since we have given that

Probability of those who attend regularly receive A's in the class = 35%

Probability of those who do not regularly receive A's in the class = 5%

Probability of students who attend class regularly = 65%

We need to find the probability that she attended class regularly given that she receives an A grade.

Let E be the event of students who attend regularly.

P(E) = 0.65

And P(E') = 1-0.65 = 0.35

Let A be the event who attend receive A in the class.

So, P(A|E) = 0.35

P(A|E') = 0.05

So, According to question, we have given that

[tex]P(E|A)=\dfrac{P(E)P(A|E)}{P(E)P(A|E)+P(E')P(A|E')}\\\\P(E|A)=\dfrac{0.65\times 0.35}{0.65\times 0.35+0.35\times 0.05}\\\\P(E|A)=\dfrac{0.2275}{0.2275+0.0175}=\dfrac{0.2275}{0.245}=0.9286[/tex]

Hence, Probability that she attended class regularly given that she receives A grade is 0.9286.

Final answer:

The probability that a student attended class regularly given they received an A is approximately 0.9286, or 92.86% when rounded to four decimal places, calculated using Bayes' theorem.

Explanation:

To solve the problem, we need to calculate the conditional probability that a student attended class regularly given they received an A grade. To do this, we'll use Bayes' theorem, which allows us to reverse conditional probabilities.

Let's denote Attendance as the event that a student attends class regularly and A as the event of a student receiving an A grade. According to the question:

The overall probability of receiving an A, P(A), is computed as follows:

P(A) = P(A|Attendance) × P(Attendance) + P(A|Not Attendance) × P(Not Attendance)
    = 0.35 × 0.65 + 0.05 × (1 - 0.65)
    = 0.2275 + 0.0175
    = 0.2450

Now we use Bayes' theorem to find P(Attendance|A), the probability of attendance given an A:

P(Attendance|A) = (P(A|Attendance) × P(Attendance)) / P(A)
       = (0.35 × 0.65) / 0.245
       = 0.2275 / 0.245
       ≈ 0.9286

Therefore, the probability that a student attended class regularly given that they received an A grade is approximately 0.9286, or 92.86% when rounded to four decimal places.

Answer:

a.[tex]w(t)=-12e^{2t}[/tex]

b.[tex] y(t)=-\frac{9}{2}e^{7t}-\frac{5}{2}e^{-5t}[/tex]

Step-by-step explanation:

We have a differential equation

y''-2 y'-35 y=0

Auxillary equation

[tex](D^2-2D-35)=0[/tex]

By factorization method we are  finding the solution

[tex]D^2-7D+5D-35=0[/tex]

[tex](D-7)(D+5)=0[/tex]

Substitute each factor equal to zero

D-7=0  and D+5=0

D=7  and D=-5

Therefore ,

General solution is

[tex]y(x)=C_1e^{7t}+C_2e^{-5t}[/tex]

Let [tex]y_1=e^{7t} \;and \;y_2=e^{-5t}[/tex]

We have to find Wronskian

[tex]w(t)=\begin{vmatrix}y_1&y_2\\y'_1&y'_2\end{vmatrix}[/tex]

Substitute values then we get

[tex]w(t)=\begin{vmatrix}e^{7t}&e^{-5t}\\7e^{7t}&-5e^{-5t}\end{vmatrix}[/tex]

[tex]w(t)=-5e^{7t}\cdot e^{-5t}-7e^{7t}\cdot e^{-5t}=-5e^{7t-5t}-7e^[7t-5t}[/tex]

[tex]w(t)=-5e^{2t}-7e^{2t}=-12e^{2t}[/tex]

a.[tex]w(t)=-12e^{2t}[/tex]

We are given that y(0)=-7 and y'(0)=23

Substitute the value in general solution the we get

[tex]y(0)=C_1+C_2[/tex]

[tex]C_1+C_2=-7[/tex]....(equation I)

[tex]y'(t)=7C_1e^{7t}-5C_2e^{-5t}[/tex]

[tex]y'(0)=7C_1-5C_2[/tex]

[tex]7C_1-5C_2=23[/tex]......(equation II)

Equation I is multiply by 5 then we subtract equation II from equation I

Using elimination method we eliminate[tex] C_1[/tex]

Then we get [tex]C_2=-\frac{5}{2}[/tex]

Substitute the value of [tex] C_2 [/tex] in  I equation then we get

[tex] C_1-\frac{5}{2}=-7[/tex]

[tex] C_1=-7+\frac{5}{2}=\frac{-14+5}{2}=-\frac{9}{2}[/tex]

Hence, the general solution is

b.[tex] y(t)=-\frac{9}{2}e^{7t}-\frac{5}{2}e^{-5t}[/tex]

Final answer:

The probability of the first letter being 'A' or 'Z' in the word ARIZONA is 0.143, the probability of it being a vowel is 0.429, and for the letter 'H', which is not present in the word, the probability is 0.

Explanation:

The probability that the first letter is 'A' in a random arrangement of the letters in the word ARIZONA is simply the number of 'A's divided by the total number of letters. Since there is one 'A' out of seven letters, the probability is 1/7, which in decimal form is approximately 0.143, rounded to the nearest thousandth.

Similarly, for the letter 'Z', since there's one 'Z' in the word ARIZONA, the probability is also 1/7, which is about 0.143 when rounded to the nearest thousandth.

The probability that the first letter is a vowel (A, I, or O in ARIZONA) involves adding the probabilities of each individual vowel being the first letter. There are three vowels out of seven letters, so the probability is 3/7, which is approximately 0.429, rounded to the nearest thousandth.

Since the letter 'H' is not in the word ARIZONA, the probability that the first letter is 'H' is 0.

Answer:

The population would be 781.

The population is growing with the rate of 12.50 persons/yr.

Step-by-step explanation:

Since, the formula for calculating the population, increasing with a rate per period,

[tex]A=P(1+r)^{n}[/tex]

Where, P is the initial population,

r is the rate per period,

n is the number of period,

t is the total years,

Here, P = 500, r = 25 % = 0.25, n = 2 ( the number of '10 year period' of in 20 years is 2 )

Hence, the population in 20 years would be,

[tex]A=500(1+0.25)^2=500(1.25)^2=781.25\approx 781[/tex]

Now, the rate of increasing per 10 year is 25 %,

⇒ The rate of increasing per year is 2.5 %,

Thus, the growing people per year = 2.5 % of 500 = 0.025 × 500 = 12.50

Hence, the population is growing at 12.50 person per year.

I suppose

[tex]H=\mathrm{span}\{10x^2+4x-1,3x-4x^2+3,5x^2+x-1\}[/tex]

The vectors that span [tex]H[/tex] form a basis for [tex]P_2[/tex] if they are (1) linearly independent and (2) any vector in [tex]P_2[/tex] can be expressed as a linear combination of those vectors (i.e. they span [tex]P_2[/tex]).

Compute the Wronskian determinant:

[tex]\begin{vmatrix}10x^2+4x-1&3x-4x^2+3&5x^2+x-1\\20x+4&3-8x&10x+1\\20&-8&10\end{vmatrix}=-6\neq0[/tex]

The determinant is non-zero, so the vectors are linearly independent. For this reason, we also know the dimension of [tex]H[/tex] is 3.

Write an arbitrary vector in [tex]P_2[/tex] as [tex]ax^2+bx+c[/tex]. Then the given vectors span [tex]P_2[/tex] if there is always a choice of scalars [tex]k_1,k_2,k_3[/tex] such that

[tex]k_1(10x^2+4x-1)+k_2(3x-4x^2+3)+k_3(5x^2+x-1)=ax^2+bx+c[/tex]

which is equivalent to the system

[tex]\begin{bmatrix}10&-4&5\\4&3&1\\-1&3&-1\end{bmatrix}\begin{bmatrix}k_1\\k_2\\k_3\end{bmatrix}=\begin{bmatrix}a\\b\\c\end{bmatrix}[/tex]

The coefficient matrix is non-singular, so it has an inverse. Multiplying both sides by that inverse gives

[tex]\begin{bmatrix}k_1\\k_2\\k_3\end{bmatrix}=\begin{bmatrix}-\dfrac{6a-11b+19c}3\\\dfrac{3a-5b+2c}3\\\dfrac{15a-26b+46c}3\end{bmatrix}[/tex]

so the vectors do span [tex]P_2[/tex].

The vectors comprising [tex]H[/tex] form a basis for it because they are linearly independent.

To determine if a set of polynomials forms a basis for P2, they need to be linearly independent and span the vector space P2. If the only solution to a homogeneous system of equations is trivial (all coefficients equal zero), they are linearly independent. Whether they span P2 or not depends on if any polynomial of degree 2 or less can be expressed as a linear combination of these polynomials.

In order to determine if the set of polynomials {10x2+4x, 3x-4x2+3, 5x2+x} forms a basis for P2, we need to prove two properties: they should be linearly independent and they should span the vector space P2.

Linear independence means that none of the polynomials in the given set can be expressed as a linear combination of the others. The simplest way to prove this is to set up a system of equations called a homogeneous system, and solve for the coefficients. If the only solution to this system is the trivial solution (where all coefficients equal zero), then they are linearly independent.

Spanning means that any polynomial of degree 2 or less can be expressed as a linear combination of these polynomials.

So, depending on the outcome of checking those two properties, we can determine if the given set of polynomials is a basis for P2 or not.

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