Find the volumes of the solids generated by revolving the triangle with vertices (2, 2)​, (2, 6)​, and (5, 6) about ​a) the​ x-axis, ​b) the​ y-axis, ​c) the line x=7​, and ​d) the line y=2.

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.

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]

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:

  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:

1) [tex]\frac{7}{20}[/tex]

2) [tex]1\frac{1}{12}[/tex]

3) [tex]\frac{7}{8}[/tex]

Step-by-step explanation:

1) Given problem,

[tex]\frac{3}{5}-\frac{1}{4}[/tex]

LCM(5,4) = 20,

[tex]\frac{3}{5}=\frac{3\times 4}{5\times 4}=\frac{12}{20}[/tex]

[tex]\frac{1}{4}=\frac{1\times 5}{4\times 5}=\frac{5}{20}[/tex]

[tex]\implies \frac{3}{5}-\frac{1}{4}=\frac{12}{20}-\frac{5}{20}=\frac{7}{20}[/tex]

2) Given problem,

[tex]\frac{5}{6}+\frac{1}{4}[/tex]

LCM(6, 4) = 12,

[tex]\frac{5}{6}=\frac{5\times 2}{6\times 2}=\frac{10}{12}[/tex]

[tex]\frac{1}{4}=\frac{1\times 3}{4\times 3}=\frac{3}{12}[/tex]

[tex]\implies \frac{5}{6}+\frac{1}{4}=\frac{10}{12}+\frac{3}{12}=\frac{13}{12}=1\frac{1}{12}[/tex]

3) ∵ Pizza ate by Manny = [tex]\frac{1}{4}[/tex]

Pizza ate by Frank = [tex]\farc{5}{8}[/tex]

∴ Total pizza eaten = [tex]\frac{1}{4}+\frac{5}{8}[/tex]

LCM(4, 8) = 8,

[tex]\frac{1}{4}=\frac{2}{8}[/tex]

Thus, total pizza eaten =  [tex]\frac{2}{8}+\frac{5}{8}=\frac{7}{8}[/tex]

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:

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: 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:

  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: (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)

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]

a)

The probability that both televisions work is:  0.42

b)

The probability at least one of the two televisions does not​ work is:

                          0.5833

There are a total of 9 televisions.

It is given that:

Three of the televisions are defective.

This means that the number of televisions which are non-defective are:

          9-3=6

a)

The probability that both televisions work is calculated by:

[tex]=\dfrac{6_C_2}{9_C_2}[/tex]

( Since 6 televisions are in working conditions and out of these 6 2 are to be selected.

and the total outcome is the selection of 2 televisions from a total of 9 televisions)

Hence, we get:

[tex]=\dfrac{\dfrac{6!}{2!\times (6-2)!}}{\dfrac{9!}{2!\times (9-2)!}}\\\\\\=\dfrac{\dfrac{6!}{2!\times 4!}}{\dfrac{9!}{2!\times 7!}}\\\\\\=\dfrac{5}{12}\\\\\\=0.42[/tex]

b)

The probability at least one of the two televisions does not​ work:

Is equal to the probability that one does not work+probability both do not work.

Probability one does not work is calculated by:

[tex]=\dfrac{3_C_1\times 6_C_1}{9_C_2}\\\\\\=\dfrac{\dfrac{3!}{1!\times (3-1)!}\times \dfrac{6!}{1!\times (6-1)!}}{\dfrac{9!}{2!\times (9-2)!}}\\\\\\=\dfrac{3\times 6}{36}\\\\\\=\dfrac{1}{2}\\\\\\=0.5[/tex]

and the probability both do not work is:

[tex]=\dfrac{3_C_2}{9_C_2}\\\\\\=\dfrac{1}{12}\\\\\\=0.0833[/tex]

Hence, Probability that atleast does not work is:

             0.5+0.0833=0.5833

To find the probability that both televisions work, use the combination formula to determine the number of ways to select 2 working televisions out of the total number of televisions. Divide this number by the total number of ways to select 2 televisions.

To find the probability that both televisions work, we need to first determine the number of ways we can select 2 televisions out of the 9 available. This can be done using the combination formula, which is C(n, r) = n!/(r!(n-r)!), where n is the total number of items and r is the number of items being selected. In this case, n = 9 and r = 2.

Now we need to determine the number of ways we can select 2 working televisions out of the 6 working televisions. Again, we can use the combination formula with n = 6 and r = 2.

The final step is to divide the number of ways we can select 2 working televisions by the total number of ways we can select 2 televisions to get the probability that both televisions work.

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Answer:

  C).  sign chart; zeroes

Step-by-step explanation:

A function potentially changes sign at each of its zeros and vertical asymptotes. So, to fill out a sign chart, you need to determine what the sign is on either side of each of these points. You can do that using test numbers, or you can do it by understanding the nature of the zero or asymptote.

Examples:

f1(x) = (x -3) . . . . changes sign at the zero x=3. Is positive for x > 3, negative for x < 3.

f2(x) = (x -4)^2 . . . . does not change sign at the zero x=4. It is positive for any x ≠ 4. This will be true for any even-degree binomial factor.

f3(x) = 1/(x+2) . . . . has a vertical asymptote at x=-2. It changes sign there because the denominator changes sign there.

f4(x) = 1/(x+3)^2 . . . . has a vertical asymptote at x=-3. It does not change sign there because the denominator is of even degree and does not change sign there.

Answer:  The correct option is (1) Dependent.

Step-by-step explanation:  For two events, we are given the following values of the probabilities :

P(A ∩ B) = 0.20,   P(A) = 0.49   and    P(B) = 0.41.

We are to check whether the events A and B are independent or dependent.

We know that

the two events C and D are said to be independent if the probabilities of their intersection is equal to the product of their probabilities.

That is,  P(C ∩ D) = P(C) × P(D).

For the given two events A and B, we have

[tex]P(A)\times P(B)=0.49\times0.41=0.2009\neq P(A\cap B)=0.20\\\\\Rightarrow P(A\cap B)\neq P(A)\times P(B).[/tex]

Therefore, the probabilities of the intersection of two events A and B is NOT equal to the product of the probabilities of the two events.

Thus, the events A and B are NOT independent. They are dependent events.

Option (1) is CORRECT.

Answer:

Step-by-step explanation:

.5

Answer is provided in image attached.

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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Answer:

The closest conversion would be C. 30 ml equals 1 fluid ounce , it is only off by 0.43 ml

Step-by-step explanation:

Great question, it is always good to ask away in order to get rid of any doubts you may be having.

The metric system is a decimal system of measurement while the household system is a system of measurement usually found with kitchen utensils. The correct conversions are the following.

5 ml equals 0.33814 tablespoon

15 ml equals 3.04326 teaspoon

29.5735 ml equals 1 fluid ounce

236.588 ml equals 1 measuring cup

So the closest conversion would be C. 30 ml equals 1 fluid ounce , it is only off by 0.43 ml

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

The correct conversion between the metric and household system provided in the choices is 30ml equals 1 fluid ounce. However, 5ml is equivalent to 1 teaspoon, 15 ml to 1 tablespoon, and 250 ml to 1 measuring cup.

The correct conversion from the metric system to the household system among the options given is C. 30 ml equals 1 fluid ounce. The rationale behind this is that 30 ml is universally accepted as being equal to 1 fluid ounce in the household system.

Option A, B and, D are incorrect conversions. More accurate conversions would be: A. 5 ml equals 1 teaspoon; B. 15 ml equals 1 tablespoon; D. 250 ml equals 1 measuring cup.

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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:  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: 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:x=-3,-2,3

Step-by-step explanation:

Given equation of polynomial is

[tex]x^{3}+2x^2-9x-18=0[/tex]

taking [tex]x^3[/tex] and -9x together and remaining together we get

[tex]x^3-9x+2x^2-18=0[/tex]

[tex]x\left ( x^2-9\right )+2\left ( x^2-9\right )[/tex]

[tex]x\left ( \left ( x+3\right )\left ( x-3\right )\right )+2\left ( \left ( x+3\right )\left ( x-3\right )\right )[/tex]

[tex]taking \left ( x+3\right )\left ( x-3\right ) as common[/tex]

[tex]\left ( x+2\right )\left ( x+3\right )\left ( x-3\right )=0[/tex]

therefore

x=-3,-2,3

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: 1.64

Step-by-step explanation:

80% = 50

20% = 12.5

100% = 62.5

38% = 1.64

Answer: The solution is,

[tex]x_1\approx 0.876[/tex]

[tex]x_2\approx 0.419[/tex]

[tex]x_3\approx 0.574[/tex]

Step-by-step explanation:

Given equations are,

[tex]8x_1 + x_2 + x_3 = 8[/tex]

[tex]2x_1 + 4x_2 + x_3 = 4[/tex]

[tex]x_1 + 3x_2 + 5x_3 = 5[/tex],

From the above equations,

[tex]x_1=\frac{1}{8}(8-x_2-x_3)[/tex]

[tex]x_2=\frac{1}{4}(4-2x_1-x_3)[/tex]

[tex]x_3=\frac{1}{5}(5-x_1-3x_2)[/tex]

First approximation,

[tex]x_1(1)=\frac{1}{8}(8-(0)-(0))=1[/tex]

[tex]x_2(1)=\frac{1}{4}(4-2(1)-(0))=0.5[/tex]

[tex]x_3(1)=\frac{1}{5}(5-1-3(0.5))=0.5[/tex]

Second approximation,

[tex]x_1(2)=\frac{1}{8}(8-(0.5)-(0.5))=0.875[/tex]

[tex]x_2(2)=\frac{1}{4}(4-2(0.875)-(0.5))=0.4375[/tex]

[tex]x_3(2)=\frac{1}{5}((0.875)-3(0.4375))=0.5625[/tex]

Third approximation,

[tex]x_1(3)=\frac{1}{8}(8-(0.4375)-(0.5625))=0.875[/tex]

[tex]x_2(3)=\frac{1}{4}(4-2(0.875)-(0.5625))=0.421875[/tex]

[tex]x_3(3)=\frac{1}{5}(5-(0.875)-3(0.421875))=0.571875[/tex]

Fourth approximation,

[tex]x_1(4)=\frac{1}{8}(8-(0.421875)-(0.571875))=0.875781[/tex]

[tex]x_2(4)=\frac{1}{4}(4-2(0.875781)-(0.571875))=0.419141[/tex]

[tex]x_3(4)=\frac{1}{5}(5-(0.875781)-3(0.419141))=0.573359[/tex]

Fifth approximation,

[tex]x_1(5)=\frac{1}{8}(8-(0.419141)-(0.573359))=0.875938[/tex]

[tex]x_2(5)=\frac{1}{4}(4-2(0.875938)-(0.573359))=0.418691[/tex]

[tex]x_3(5)=\frac{1}{5}(5-(0.875938)-3(0.418691))=0.573598[/tex]

Hence, by the Gauss Seidel method the solution of the given system is,

[tex]x_1\approx 0.876[/tex]

[tex]x_2\approx 0.419[/tex]

[tex]x_3\approx 0.574[/tex]

Answer: first option.

Step-by-step explanation:

By definition, the measure of any interior angle of an equilateral triangle is 60 degrees.

Then, we can find the value of "y". This is:

[tex]2y+6=60\\\\y=\frac{54}{2}\\\\y=27[/tex]

Since the three sides of an equilateral triangle have the same length, we can find the value of "x". This is:

[tex]x+4=2x-3\\\\4+3=2x-x\\\\x=7[/tex]

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)

Answer:

x = 25

Step-by-step explanation:

Step 1: Identify the similar triangles

Triangle DQC and triangle DBR are similar

Step 2: Identify the parallel lines

QC is parallel to BR

Step 3: Find x

DQ/QB = DC/CR

40/24 = x/15

x = 25

!!

Answer: [tex]x=25[/tex]

Step-by-step explanation:

In order to calculate the value of "x", you can set up de following proportion:

[tex]\frac{BQ+QD}{QD}=\frac{RC+CD}{CD}\\\\\frac{24+40}{40}=\frac{15+x}{x}[/tex]

Now, the final step is to solve for "x" to find its value.

Therefore, its value is the following:

[tex]1.6=\frac{15+x}{x}\\\\1.6x=x+15\\\\1.6x-x=15\\\\0.6x=15\\\\x=\frac{15}{0.6}\\\\x=25[/tex]

The norm of a vector [tex]\vec x[/tex] is equal to the square root of the inner product of [tex]\vec x[/tex] with itself.

a. [tex]\|(-1,2)\|=\sqrt{\langle(-1,2),(-1,2)\rangle}=\sqrt{4(-1)^2+9(2)^2}=\sqrt{40}=2\sqrt{10}[/tex]

b. [tex]\|(3,2)\|=\sqrt{\langle(3,2),(3,2)\rangle}=\sqrt{4(3)^2+9(2)^2}=\sqrt{72}=6\sqrt2[/tex]

Answer:Find the slope of the line that passes through the points shown in the table.

The slope of the line that passes through the points in the table is

.

Step-by-step explanation:

By substitifying the given points into the objective function, we can evaluate the minimum P. The point (x, y) = (0, 0) gives the minimum value of P = 0, which is the optimal solution for this problem.

This problem is a classic example of a linear programming problem, a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. In this case, we are asked to minimize P = 3x + 15y subject to the constraints [tex]2x + 4y \leq 12, 5x + 2y \leq 10, and ,x \geq 0, y \geq 0.[/tex] In other words, we are looking for values of x and y that satisfy the constraints and result in the smallest possible value of P.

By substituting our given points into the equation for P we can compare the results. The smallest value for P corresponds to the point (x, y) = (0, 0) with P = 0. This is the optimal solution for this problem because it results in the lowest value for P while still satisfying all the constraints.

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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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